feat: Add complex multibrot fractal

src/app.rs

@@ -56,6 +56,7 @@ const KINDS: &[(FractalKind, &str)] = &[
     (FractalKind::Buffalo, "Buffalo"),
     (FractalKind::Phoenix, "Phoenix"),
     (FractalKind::Lambda, "Lambda"),
+    (FractalKind::ComplexMultibrot, "Complex Multibrot"),
 ];
 
 /// UI label for a fractal kind.
@@ -69,8 +70,8 @@ fn kind_label(kind: FractalKind) -> &'static str {
 
 /// The iteration formula for a kind, in human-readable notation (mirrors the
 /// doc comments on `FractalKind`'s variants). `power` is only used by
-/// Multibrot.
+/// Multibrot; `complex_power` only by Complex Multibrot.
-fn kind_formula(kind: FractalKind, power: u32) -> String {
+fn kind_formula(kind: FractalKind, power: u32, complex_power: (f64, f64)) -> String {
     match kind {
         FractalKind::Mandelbrot => "z = z² + c".to_string(),
         FractalKind::BurningShip => "z = (|Re(z)| + i|Im(z)|)² + c".to_string(),
@@ -81,13 +82,16 @@ fn kind_formula(kind: FractalKind, power: u32) -> String {
         FractalKind::Buffalo => "z = |Re(z²)| − i|Im(z²)| + c".to_string(),
         FractalKind::Phoenix => "z = z² + c + p·z_prev".to_string(),
         FractalKind::Lambda => "z = λ·z(1 − z)".to_string(),
+        FractalKind::ComplexMultibrot => {
+            format!("z = z^({:.3}{:+.3}i) + c", complex_power.0, complex_power.1)
+        }
     }
 }
 
 type JuliaPreset = (&'static str, f64, f64, u32, Option<(f64, f64)>);
 
 /// Nice-looking Julia constants offered as presets.
-const JULIA_PRESETS: [&[JuliaPreset]; FractalKind::Lambda as usize + 1] = [
+const JULIA_PRESETS: [&[JuliaPreset]; FractalKind::ComplexMultibrot as usize + 1] = [
     &[
         ("dendrite", -0.8, 0.156, 400, None),
         ("rabbit", -0.123, 0.745, 400, None),
@@ -106,6 +110,7 @@ const JULIA_PRESETS: [&[JuliaPreset]; FractalKind::Lambda as usize + 1] = [
         ("archipelago 2", -0.556, 0.253, 500, Some((-0.415, -0.267))),
     ],
     &[],
+    &[],
 ];
 
 type SetPreset = (
@@ -120,7 +125,7 @@ type SetPreset = (
 /// Curated beautiful locations offered as one-click presets.
 /// Each is `(name, center_re, center_im, half_height, iterations)`; the centers
 /// are decimals parsed at full precision so deep places stay sharp.
-const SET_PRESETS: [&[SetPreset]; FractalKind::Lambda as usize + 1] = [
+const SET_PRESETS: [&[SetPreset]; FractalKind::ComplexMultibrot as usize + 1] = [
     &[
         (
             "Seahorse Valley",
@@ -171,6 +176,7 @@ const SET_PRESETS: [&[SetPreset]; FractalKind::Lambda as usize + 1] = [
         Some((-0.9, -0.49)),
     )],
     &[],
+    &[],
 ];
 
 /// Parameters a reference orbit was (or will be) computed for. Used to decide
@@ -186,6 +192,7 @@ struct RequestKey {
     iter: u32,
     kind: FractalKind,
     power: u32,
+    complex_power: (f64, f64),
 }
 
 /// Shared state for an in-progress PNG export. The worker (a background thread
@@ -273,6 +280,8 @@ pub struct FractalApp {
     kind: FractalKind,
     /// Exponent for the Multibrot kind.
     power: u32,
+    /// Complex exponent for the Complex Multibrot kind (`z^power + c`).
+    complex_power: (f64, f64),
     julia_c: (f64, f64),
     /// Distortion constant `p` for the Phoenix kind (`z^2 + c + p·z_{n-1}`).
     phoenix_p: (f64, f64),
@@ -452,6 +461,7 @@ impl FractalApp {
             mode: FractalMode::Mandelbrot,
             kind: FractalKind::Mandelbrot,
             power: 3,
+            complex_power: (2.0, 0.5),
             julia_c: (-0.8, 0.156),
             phoenix_p: (-0.5, 0.0),
             lambda_l: (-0.5, 0.0),
@@ -513,6 +523,15 @@ impl FractalApp {
             if let Some(p) = cli.power {
                 self.power = p.clamp(2, 8);
             }
+            if let Some(cp) = &cli.complex_power {
+                let p: Vec<&str> = cp.split(',').collect();
+                if let (Some(Ok(re)), Some(Ok(im))) = (
+                    p.first().map(|s| s.trim().parse::<f64>()),
+                    p.get(1).map(|s| s.trim().parse::<f64>()),
+                ) {
+                    self.complex_power = (re, im);
+                }
+            }
             self.view = Self::default_view_for(self.mode, self.kind);
         }
         if let Some(jc) = cli.julia {
@@ -618,6 +637,7 @@ impl FractalApp {
             julia_c: self.julia_c,
             phoenix_p: self.phoenix_p,
             lambda_l: self.lambda_l,
+            complex_power: self.complex_power,
             color_scale: self.color_scale,
             color_offset: self.color_offset,
             palette: self.palette,
@@ -637,6 +657,7 @@ impl FractalApp {
         self.julia_c = s.julia_c;
         self.phoenix_p = s.phoenix_p;
         self.lambda_l = s.lambda_l;
+        self.complex_power = s.complex_power;
         self.color_scale = s.color_scale;
         self.color_offset = s.color_offset;
         self.palette = (s.palette as usize).min(PALETTE_NAMES.len() - 1) as u32;
@@ -688,6 +709,7 @@ impl FractalApp {
             FractalKind::Buffalo => (-0.5, -0.5, 1.5),
             FractalKind::Phoenix => (0.0, 0.0, 1.6),
             FractalKind::Lambda => (0.0, 0.0, 1.6),
+            FractalKind::ComplexMultibrot => (0.0, 0.0, 1.5),
         };
         ViewState::with_center(big_from_f64(cr, 53), big_from_f64(ci, 53), hh)
     }
@@ -704,6 +726,7 @@ impl FractalApp {
             iter: self.max_iterations,
             kind: self.kind,
             power: self.power,
+            complex_power: self.complex_power,
         }
     }
 
@@ -729,6 +752,7 @@ impl FractalApp {
             || key.iter != self.max_iterations
             || key.kind != self.kind
             || key.power != self.power
+            || key.complex_power != self.complex_power
         {
             return true;
         }
@@ -797,6 +821,7 @@ impl FractalApp {
                     power: key.power,
                     phoenix_p: key.phoenix_p,
                     lambda_l: key.lambda_l,
+                    complex_power: key.complex_power,
                 });
                 self.pending = true;
             }
@@ -816,6 +841,7 @@ impl FractalApp {
                         key.power,
                         key.phoenix_p,
                         key.lambda_l,
+                        key.complex_power,
                     )
                 } else {
                     compute_set_reference(
@@ -827,6 +853,7 @@ impl FractalApp {
                         key.power,
                         key.phoenix_p,
                         key.lambda_l,
+                        key.complex_power,
                     )
                 };
                 self.apply_reference(
@@ -881,6 +908,7 @@ impl FractalApp {
                 key.power,
                 key.phoenix_p,
                 key.lambda_l,
+                key.complex_power,
             )
         } else {
             compute_set_reference(
@@ -892,6 +920,7 @@ impl FractalApp {
                 key.power,
                 key.phoenix_p,
                 key.lambda_l,
+                key.complex_power,
             )
         };
         self.apply_reference(
@@ -921,6 +950,7 @@ impl FractalApp {
             dc_offset: self.dc_offset(),
             phoenix_p: [self.phoenix_p.0 as f32, self.phoenix_p.1 as f32],
             lambda_l: [self.lambda_l.0 as f32, self.lambda_l.1 as f32],
+            complex_power: [self.complex_power.0 as f32, self.complex_power.1 as f32],
             de_coloring: (self.de_coloring | self.shadow) as u32,
             shadow: self.shadow as u32,
             _pad: [0; _],
@@ -941,6 +971,7 @@ impl FractalApp {
             aspect: aspect as f32,
             phoenix_p: [self.phoenix_p.0 as f32, self.phoenix_p.1 as f32],
             lambda_l: [self.lambda_l.0 as f32, self.lambda_l.1 as f32],
+            complex_power: [self.complex_power.0 as f32, self.complex_power.1 as f32],
             bailout_sq: BAILOUT_SQ,
             kind: self.kind as u32,
             power: self.power,
@@ -954,7 +985,7 @@ impl FractalApp {
             height: 0,          // set by the callback from size_px
             total_samples: 0.0, // tracked by the renderer across frames
             palette: self.buddha_palette,
-            _pad: [0; 3],
+            _pad0: 0,
         }
     }
 
@@ -1246,6 +1277,12 @@ impl FractalApp {
                         self.lambda_l.0, self.lambda_l.1
                     ));
                 }
+                if self.kind == FractalKind::ComplexMultibrot {
+                    ui.label(format!(
+                        "power = {:.6} {:+.6}i",
+                        self.complex_power.0, self.complex_power.1
+                    ));
+                }
                 ui.separator();
 
                 ui.label(self.kind.description());
@@ -1270,7 +1307,8 @@ impl FractalApp {
                         ui.label(
                             "A deep-zoom fractal explorer. It renders the Mandelbrot set \
                              and several related fractals (Burning Ship, Tricorn, \
-                             Multibrot, Celtic, Perpendicular, Buffalo, Phoenix, Lambda).",
+                             Multibrot, Complex Multibrot, Celtic, Perpendicular, Buffalo, \
+                             Phoenix, Lambda).",
                         );
                         ui.add_space(4.0);
                         ui.label(
@@ -1511,6 +1549,22 @@ impl FractalApp {
                 ui.label("i");
             });
         }
+        if self.kind == FractalKind::ComplexMultibrot {
+            ui.horizontal(|ui| {
+                ui.label("power =");
+                ui.add(
+                    egui::DragValue::new(&mut self.complex_power.0)
+                        .speed(0.01)
+                        .range(-8.0..=8.0),
+                );
+                ui.add(
+                    egui::DragValue::new(&mut self.complex_power.1)
+                        .speed(0.01)
+                        .range(-8.0..=8.0),
+                );
+                ui.label("i");
+            });
+        }
         if self.kind != prev_kind {
             self.view = Self::default_view_for(self.mode, self.kind);
         }

src/cli.rs

@@ -17,6 +17,10 @@ pub struct Cli {
     #[arg(long)]
     pub power: Option<u32>,
 
+    /// Complex exponent for the Complex Multibrot kind (z -> z^power + c).
+    #[arg(long, value_name = "RE,IM")]
+    pub complex_power: Option<String>,
+
     /// Start in Julia mode with this seed constant.
     #[arg(long, value_name = "RE,IM")]
     pub julia: Option<String>,
@@ -79,6 +83,8 @@ pub enum KindArg {
     Buffalo,
     Phoenix,
     Lambda,
+    #[value(alias = "cmulti")]
+    ComplexMultibrot,
 }
 
 impl From<KindArg> for FractalKind {
@@ -93,6 +99,7 @@ impl From<KindArg> for FractalKind {
             KindArg::Buffalo => FractalKind::Buffalo,
             KindArg::Phoenix => FractalKind::Phoenix,
             KindArg::Lambda => FractalKind::Lambda,
+            KindArg::ComplexMultibrot => FractalKind::ComplexMultibrot,
         }
     }
 }

src/fractal/buddhabrot.rs

@@ -46,7 +46,11 @@ pub struct BuddhabrotUniforms {
     /// (yellow core, blue halo), 2 = grayscale. Display-only, like `exposure`
     /// — excluded from `ContentKey` so changing it doesn't reset accumulation.
     pub palette: u32,
-    pub _pad: [u32; 3],
+    /// Padding so `complex_power` (a vec2, 8-byte aligned in the shader)
+    /// starts on an 8-byte boundary.
+    pub _pad0: u32,
+    /// Complex exponent for the Complex Multibrot kind; ignored by other kinds.
+    pub complex_power: [f32; 2],
 }
 
 /// The subset of `BuddhabrotUniforms` that determines the *content* of the
@@ -62,6 +66,7 @@ struct ContentKey {
     bailout_sq: f32,
     kind: u32,
     power: u32,
+    complex_power: [f32; 2],
     r_cap: u32,
     g_cap: u32,
     b_cap: u32,
@@ -78,6 +83,7 @@ impl From<&BuddhabrotUniforms> for ContentKey {
             bailout_sq: u.bailout_sq,
             kind: u.kind,
             power: u.power,
+            complex_power: u.complex_power,
             r_cap: u.r_cap,
             g_cap: u.g_cap,
             b_cap: u.b_cap,

src/fractal/reference.rs

@@ -35,6 +35,9 @@ pub enum FractalKind {
     Phoenix = 7,
     /// `z -> lambda·z(1 - z)` (logistic map).
     Lambda = 8,
+    /// `z -> z^power + c`, where `power` is a complex constant (the
+    /// `complex_power` argument), via the principal branch `z^p = exp(p·ln z)`.
+    ComplexMultibrot = 9,
 }
 
 impl FractalKind {
@@ -53,6 +56,7 @@ impl FractalKind {
             FractalKind::Buffalo => "",
             FractalKind::Phoenix => "",
             FractalKind::Lambda => "",
+            FractalKind::ComplexMultibrot => "Like Multibrot, but the exponent itself is a complex number instead of a plain integer, via z^p = exp(p·ln z).",
         }
     }
 }
@@ -77,6 +81,7 @@ pub fn compute_reference(
     power: u32,
     phoenix_p: (f64, f64),
     lambda_l: (f64, f64),
+    complex_power: (f64, f64),
 ) -> Vec<[f32; 2]> {
     let cr = c_re.clone().with_precision(precision).value();
     let ci = c_im.clone().with_precision(precision).value();
@@ -92,6 +97,9 @@ pub fn compute_reference(
     // Lambda distortion constant `l` (a small fixed complex number).
     let lr = big_from_f64(lambda_l.0, precision);
     let li = big_from_f64(lambda_l.1, precision);
+    // Complex Multibrot exponent (a fixed complex number).
+    let cpow_re = big_from_f64(complex_power.0, precision);
+    let cpow_im = big_from_f64(complex_power.1, precision);
 
     let mut points: Vec<[f32; 2]> = Vec::with_capacity(max_iter as usize + 1);
 
@@ -162,6 +170,10 @@ pub fn compute_reference(
                 let lzi = &lr * &zi + &li * &zr;
                 (&lzr * &re2 - &lzi * &im2, re2 * lzi + lzr * im2)
             }
+            FractalKind::ComplexMultibrot => {
+                let (pr, pi) = complex_pow_complex(&zr, &zi, &cpow_re, &cpow_im, precision);
+                (pr + &cr, pi + &ci)
+            }
         };
 
         // Shift the previous iterate (only the Phoenix arm reads it).
@@ -198,6 +210,32 @@ fn complex_pow(zr: &Big, zi: &Big, power: u32, precision: usize) -> (Big, Big) {
     (rr, ri)
 }
 
+/// `true` if `x` is (numerically) zero. The f64 check is exact for a true
+/// zero; only matters here to special-case `ln(0)`.
+fn is_big_zero(x: &Big) -> bool {
+    x.to_f64().value() == 0.0
+}
+
+/// `(zr + i zi)^(pr + i pi)` for a complex exponent, via the principal branch
+/// `z^p = exp(p·ln z)` where `ln z = ln|z| + i·arg(z)`. Used by
+/// `ComplexMultibrot`; must be kept in sync with the shader's `cpow`.
+/// `z = 0` is special-cased to `0` (the formula's `ln(0)` would otherwise
+/// panic; this is the correct limit for the `Re(p) > 0` region the UI
+/// exposes).
+fn complex_pow_complex(zr: &Big, zi: &Big, pr: &Big, pi: &Big, precision: usize) -> (Big, Big) {
+    if is_big_zero(zr) && is_big_zero(zi) {
+        return (big_zero(precision), big_zero(precision));
+    }
+    let r2 = &zr.sqr() + &zi.sqr();
+    let ln_r = r2.ln() >> 1; // 0.5 * ln(r2) = ln(sqrt(r2)); exact halving.
+    let theta = zi.atan2(zr);
+    let exp_re = (pr * &ln_r - pi * &theta).with_precision(precision).value();
+    let exp_im = (pr * &theta + pi * &ln_r).with_precision(precision).value();
+    let mag = exp_re.exp();
+    let (sin_a, cos_a) = exp_im.sin_cos();
+    (&mag * &cos_a, &mag * &sin_a)
+}
+
 /// Convenience: parameter-plane ("Mandelbrot-set") reference (`z0 = 0`,
 /// `c = center`) for any `kind`.
 #[allow(clippy::too_many_arguments)]
@@ -210,10 +248,21 @@ pub fn compute_set_reference(
     power: u32,
     phoenix_p: (f64, f64),
     lambda_l: (f64, f64),
+    complex_power: (f64, f64),
 ) -> Vec<[f32; 2]> {
     let zero = big_zero(precision);
     compute_reference(
-        &zero, &zero, center_re, center_im, max_iter, precision, kind, power, phoenix_p, lambda_l,
+        &zero,
+        &zero,
+        center_re,
+        center_im,
+        max_iter,
+        precision,
+        kind,
+        power,
+        phoenix_p,
+        lambda_l,
+        complex_power,
     )
 }
 
@@ -236,6 +285,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         // Independent naive f64 orbit.
@@ -275,6 +325,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
         assert_eq!(points.len(), 501, "interior orbit should not escape");
     }
@@ -293,6 +344,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (-1.75_f64, -0.03_f64);
@@ -322,6 +374,7 @@ mod tests {
             3,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (0.3_f64, 0.2_f64);
@@ -357,6 +410,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (mut zr, mut zi) = (0.15_f64, -0.1_f64);
@@ -386,6 +440,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (-0.6_f64, 0.4_f64);
@@ -416,6 +471,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (-0.7_f64, -0.2_f64);
@@ -446,6 +502,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (-1.2_f64, -0.35_f64);
@@ -468,8 +525,17 @@ mod tests {
         let cr = Big::try_from(0.5667_f64).unwrap();
         let ci = Big::try_from(0.0_f64).unwrap();
         let p = (-0.5_f64, 0.0_f64);
-        let points =
+        let points = compute_set_reference(
-            compute_set_reference(&cr, &ci, 60, 200, FractalKind::Phoenix, 2, p, (0.0, 0.0));
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::Phoenix,
+            2,
+            p,
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
 
         let (c_re, c_im) = (0.5667_f64, 0.0_f64);
         let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
@@ -489,4 +555,50 @@ mod tests {
             zi = nzi;
         }
     }
+
+    /// Complex Multibrot (power 2.5 + 0.3i) reference matches a naive f64
+    /// iteration of `z^p = exp(p·ln z)`.
+    #[test]
+    fn complex_multibrot_reference_matches_naive_f64() {
+        let cr = Big::try_from(0.1_f64).unwrap();
+        let ci = Big::try_from(-0.2_f64).unwrap();
+        let power = (2.5_f64, 0.3_f64);
+        let points = compute_set_reference(
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::ComplexMultibrot,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+            power,
+        );
+
+        // Naive f64 complex power via z^p = exp(p * ln z), ln z = ln|z| + i*arg(z).
+        fn naive_cpow(zr: f64, zi: f64, pr: f64, pi: f64) -> (f64, f64) {
+            if zr == 0.0 && zi == 0.0 {
+                return (0.0, 0.0);
+            }
+            let ln_r = 0.5 * (zr * zr + zi * zi).ln();
+            let theta = zi.atan2(zr);
+            let exp_re = pr * ln_r - pi * theta;
+            let exp_im = pr * theta + pi * ln_r;
+            let mag = exp_re.exp();
+            (mag * exp_im.cos(), mag * exp_im.sin())
+        }
+
+        let (c_re, c_im) = (0.1_f64, -0.2_f64);
+        let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
+        for point in &points {
+            let tol = 1e-4 * (1.0 + zr.abs().max(zi.abs()));
+            assert!((point[0] as f64 - zr).abs() < tol, "re: {point:?} vs {zr}");
+            assert!((point[1] as f64 - zi).abs() < tol, "im: {point:?} vs {zi}");
+            let (pr, pi) = naive_cpow(zr, zi, power.0, power.1);
+            let nzr = pr + c_re;
+            let nzi = pi + c_im;
+            zr = nzr;
+            zi = nzi;
+        }
+    }
 }

src/fractal/renderer.rs

@@ -37,6 +37,7 @@ fn geom_differs(a: &Uniforms, b: &Uniforms) -> bool {
         || a.aa_level != b.aa_level
         || a.kind != b.kind
         || a.power != b.power
+        || a.complex_power != b.complex_power
         || a.dc_offset != b.dc_offset
         || a.phoenix_p != b.phoenix_p
         || a.de_coloring != b.de_coloring
@@ -87,6 +88,9 @@ pub struct Uniforms {
     /// Distortion constant `l` for the Lambda map (`l·z(1 - z)`);
     /// ignored by other kinds.
     pub lambda_l: [f32; 2],
+    /// Complex exponent for the Complex Multibrot kind (`z^power + c`);
+    /// ignored by other kinds.
+    pub complex_power: [f32; 2],
     /// 0 = escape-time coloring, 1 = distance-estimation shading.
     pub de_coloring: u32,
     // 0 = classic colors, 1 = shadows

src/fractal/share.rs

@@ -25,6 +25,8 @@ pub struct ShareState {
     pub phoenix_p: (f64, f64),
     /// Distortion constant for the Lambda kind (ignored by others).
     pub lambda_l: (f64, f64),
+    /// Complex exponent for the Complex Multibrot kind (ignored by others).
+    pub complex_power: (f64, f64),
     pub color_scale: f32,
     pub color_offset: f32,
     /// Palette index (`palette_id` in the shader).
@@ -49,6 +51,7 @@ impl ShareState {
                 FractalKind::Buffalo => "buffalo",
                 FractalKind::Phoenix => "phoenix",
                 FractalKind::Lambda => "lambda",
+                FractalKind::ComplexMultibrot => "cmulti",
             }
         ));
         s.push_str(&format!("&pw={}", self.power));
@@ -60,8 +63,12 @@ impl ShareState {
         s.push_str(&format!("&px={}&py={}", self.phoenix_p.0, self.phoenix_p.1));
         s.push_str(&format!("&lx={}&ly={}", self.lambda_l.0, self.lambda_l.1));
         s.push_str(&format!(
-            "&cs={}&co={}&pal={}",
+            "&cpr={}&cpi={}",
-            self.color_scale, self.color_offset, self.palette
+            self.complex_power.0, self.complex_power.1
+        ));
+        s.push_str(&format!(
+            "&cs={}&co={}&pal={}&spal={}",
+            self.color_scale, self.color_offset, self.palette, self.shadow_palette
         ));
         s
     }
@@ -89,6 +96,7 @@ impl ShareState {
                     "buffalo" => FractalKind::Buffalo,
                     "phoenix" => FractalKind::Phoenix,
                     "lambda" => FractalKind::Lambda,
+                    "cmulti" => FractalKind::ComplexMultibrot,
                     _ => FractalKind::Mandelbrot,
                 })
                 .unwrap_or(FractalKind::Mandelbrot),
@@ -109,6 +117,10 @@ impl ShareState {
                 map.get("lx").and_then(|s| s.parse().ok()).unwrap_or(-0.5),
                 map.get("ly").and_then(|s| s.parse().ok()).unwrap_or(0.0),
             ),
+            complex_power: (
+                map.get("cpr").and_then(|s| s.parse().ok()).unwrap_or(2.0),
+                map.get("cpi").and_then(|s| s.parse().ok()).unwrap_or(0.0),
+            ),
             color_scale: map.get("cs").and_then(|s| s.parse().ok()).unwrap_or(0.02),
             color_offset: map.get("co").and_then(|s| s.parse().ok()).unwrap_or(0.0),
             palette: map.get("pal").and_then(|s| s.parse().ok()).unwrap_or(0),
@@ -134,6 +146,7 @@ mod tests {
             julia_c: (-0.123, 0.745),
             phoenix_p: (-0.5, 0.1),
             lambda_l: (-0.5, 0.0),
+            complex_power: (2.5, 0.3),
             color_scale: 0.02,
             color_offset: 0.25,
             palette: 3,
@@ -149,6 +162,7 @@ mod tests {
         assert_eq!(d.iterations, s.iterations);
         assert_eq!(d.julia_c, s.julia_c);
         assert_eq!(d.phoenix_p, s.phoenix_p);
+        assert_eq!(d.complex_power, s.complex_power);
         assert_eq!(d.palette, s.palette);
         assert_eq!(d.shadow_palette, s.shadow_palette);
     }

src/shaders/buddhabrot.wgsl

@@ -47,14 +47,15 @@ struct Uniforms {
     // Tonemap colour style: 0 = classic (R/G/B = raw caps), 1 = nebula
     // (yellow core, blue halo), 2 = grayscale.
     palette: u32,
-    // Padding to a 16-byte multiple. NOT vec3<u32> — that type aligns to 16
+    // Padding so `complex_power` (a vec2, 8-byte aligned) starts on an
-    // bytes in WGSL (unlike Rust's `[u32; 3]`, which aligns to 4), which
+    // 8-byte boundary. NOT vec3<u32> — that type aligns to 16 bytes in WGSL
-    // silently added 32 bytes instead of 16 and mismatched the Rust struct's
+    // (unlike Rust's `[u32; 3]`, which aligns to 4), which silently added 32
-    // size (a wgpu validation error at dispatch time: "size 96 where the
+    // bytes instead of 16 and mismatched the Rust struct's size (a wgpu
-    // shader expects 112").
+    // validation error at dispatch time: "size 96 where the shader expects
+    // 112").
     _pad0: u32,
-    _pad1: u32,
+    // Complex exponent for the Complex Multibrot kind; unused by other kinds.
-    _pad2: u32,
+    complex_power: vec2<f32>,
 };
 
 const PALETTE_NEBULA: u32 = 0u;
@@ -70,6 +71,7 @@ const KIND_PERPENDICULAR: u32 = 5u;
 const KIND_BUFFALO: u32 = 6u;
 const KIND_PHOENIX: u32 = 7u;
 const KIND_LAMBDA: u32 = 8u;
+const KIND_COMPLEX_MULTIBROT: u32 = 9u;
 
 @group(0) @binding(0) var<uniform> u: Uniforms;
 // Compute pass: read-write atomic histogram (3 planes of width*height, R/G/B).
@@ -103,6 +105,21 @@ fn complex_pow(z: vec2<f32>, p: u32) -> vec2<f32> {
     return r;
 }
 
+// z^p for a complex exponent p, via the principal branch z^p = exp(p * ln z),
+// ln z = ln|z| + i*arg(z). z = 0 maps to 0 (the correct limit for the
+// Re(p) > 0 region the UI exposes; ln(0) would otherwise be -inf).
+fn cpow(z: vec2<f32>, p: vec2<f32>) -> vec2<f32> {
+    let r2 = dot(z, z);
+    if r2 < 1e-30 {
+        return vec2<f32>(0.0, 0.0);
+    }
+    let ln_r = 0.5 * log(r2);
+    let theta = atan2(z.y, z.x);
+    let mag = exp(p.x * ln_r - p.y * theta);
+    let ang = p.x * theta + p.y * ln_r;
+    return mag * vec2<f32>(cos(ang), sin(ang));
+}
+
 // One iteration step z_n -> z_{n+1} for the current kind. `zp` is the
 // previous iterate (z_{n-1}), used only by the Phoenix two-term recurrence.
 // Must match `FractalKind` in reference.rs (the direct, non-perturbative form
@@ -126,6 +143,8 @@ fn advance(z: vec2<f32>, zp: vec2<f32>, c: vec2<f32>) -> vec2<f32> {
     } else if u.kind == KIND_LAMBDA {
         // l * z * (1 - z); c is unused (see file doc comment above).
         return cmul(u.lambda_l, cmul(z, vec2<f32>(1.0 - z.x, -z.y)));
+    } else if u.kind == KIND_COMPLEX_MULTIBROT {
+        return cpow(z, u.complex_power) + c;
     }
     return vec2<f32>(z.x * z.x - z.y * z.y, 2.0 * z.x * z.y) + c; // Mandelbrot
 }

src/shaders/colorize.wgsl

@@ -27,6 +27,7 @@ struct Uniforms {
     dc_offset: vec2<f32>,
     phoenix_p: vec2<f32>,
     lambda_l: vec2<f32>,
+    complex_power: vec2<f32>,
     de_coloring: u32,
     shadow: u32,
 };

src/shaders/mandelbrot.wgsl

@@ -34,6 +34,9 @@ struct Uniforms {
     // Distortion constant l for the Lambda map (l*z(1 - z_{n-1})); unused
     // by other kinds.
     lambda_l: vec2<f32>,
+    // Complex exponent for the Complex Multibrot kind (z^power + c); unused
+    // by other kinds.
+    complex_power: vec2<f32>,
     // 0 = escape-time coloring, 1 = distance-estimation shading.
     de_coloring: u32,
     // 0 = classic colors, 1 = shadows
@@ -49,6 +52,7 @@ const KIND_PERPENDICULAR: u32 = 5u;
 const KIND_BUFFALO: u32 = 6u;
 const KIND_PHOENIX: u32 = 7u;
 const KIND_LAMBDA: u32 = 8u;
+const KIND_COMPLEX_MULTIBROT: u32 = 9u;
 
 @group(0) @binding(0) var<uniform> u: Uniforms;
 @group(0) @binding(1) var<storage, read> ref_orbit: array<vec2<f32>>;
@@ -84,6 +88,27 @@ fn conj(a: vec2<f32>) -> vec2<f32> {
     return vec2<f32>(a.x, -a.y);
 }
 
+// Complex division a / b.
+fn cdiv(a: vec2<f32>, b: vec2<f32>) -> vec2<f32> {
+    let d = dot(b, b);
+    return vec2<f32>(a.x * b.x + a.y * b.y, a.y * b.x - a.x * b.y) / d;
+}
+
+// z^p for a complex exponent p, via the principal branch z^p = exp(p * ln z),
+// ln z = ln|z| + i*arg(z). z = 0 maps to 0 (the correct limit for the
+// Re(p) > 0 region the UI exposes; ln(0) would otherwise be -inf).
+fn cpow(z: vec2<f32>, p: vec2<f32>) -> vec2<f32> {
+    let r2 = dot(z, z);
+    if r2 < 1e-30 {
+        return vec2<f32>(0.0, 0.0);
+    }
+    let ln_r = 0.5 * log(r2);
+    let theta = atan2(z.y, z.x);
+    let mag = exp(p.x * ln_r - p.y * theta);
+    let ang = p.x * theta + p.y * ln_r;
+    return mag * vec2<f32>(cos(ang), sin(ang));
+}
+
 // |c + d| - |c|, evaluated exactly (no catastrophic cancellation even when the
 // sum crosses zero). This is what makes the Burning Ship delta correct through
 // the sign flips that happen all along the axes, where the ship's detail lives.
@@ -123,6 +148,38 @@ fn multibrot_delta(z: vec2<f32>, e: vec2<f32>, p: u32) -> vec2<f32> {
     return acc;
 }
 
+// Number of terms kept in `complex_multibrot_delta`'s series. Truncation, not
+// exactness: unlike `multibrot_delta` (a finite binomial sum for an integer
+// power), a complex power has no finite expansion, so this converges rather
+// than terminates. Fine as long as perturbation's usual invariant (|e| << |z|,
+// kept true by rebasing) holds, since each extra term is O(w^k) smaller.
+const COMPLEX_MULTIBROT_TERMS: u32 = 16u;
+
+// Perturbation delta for z -> z^p with a complex p: (Z+e)^p - Z^p.
+// = Z^p * ((1+w)^p - 1), w = e/Z, expanded as a Taylor series in w (never
+// forming 1+w, which would round tiny w away in f32 — the same reason
+// `multibrot_delta` never forms Z+e directly). Series: (1+w)^p - 1 =
+// sum_{k=1}^N C(p,k) w^k, with the complex binomial coefficient built up
+// incrementally: C(p,k) = C(p,k-1) * (p-(k-1)) / k.
+//
+// Z ~ 0 (the reference start, X_0 = 0 for Mandelbrot) makes w singular; there
+// (0+e)^p - 0^p = e^p exactly, so that case is handled directly via `cpow`.
+fn complex_multibrot_delta(z: vec2<f32>, e: vec2<f32>, p: vec2<f32>) -> vec2<f32> {
+    if dot(z, z) < 1e-20 {
+        return cpow(e, p);
+    }
+    let w = cdiv(e, z);
+    var wk = vec2<f32>(1.0, 0.0); // w^0
+    var coef = vec2<f32>(1.0, 0.0); // C(p,0)
+    var acc = vec2<f32>(0.0, 0.0);
+    for (var k: u32 = 1u; k <= COMPLEX_MULTIBROT_TERMS; k = k + 1u) {
+        coef = cdiv(cmul(coef, p - vec2<f32>(f32(k - 1u), 0.0)), vec2<f32>(f32(k), 0.0));
+        wk = cmul(wk, w);
+        acc = acc + cmul(coef, wk);
+    }
+    return cmul(cpow(z, p), acc);
+}
+
 // One perturbation step of the current fractal's delta: e -> f(Z+e) - f(Z),
 // where `z` is the reference orbit value X_m. `step_add` (dc) is added by the
 // caller. Must match `FractalKind` on the CPU side.
@@ -163,6 +220,8 @@ fn advance_delta(z: vec2<f32>, e: vec2<f32>) -> vec2<f32> {
         // Lambda map: z^{n+1} = λ·z·(1-z). Delta: e = λ·e·(1-2z-e).
         let one_minus_2z_minus_e = vec2<f32>(1.0 - 2.0 * z.x - e.x, -2.0 * z.y - e.y);
         return cmul(u.lambda_l, cmul(e, one_minus_2z_minus_e));
+    } else if u.kind == KIND_COMPLEX_MULTIBROT {
+        return complex_multibrot_delta(z, e, u.complex_power);
     }
     return 2.0 * cmul(z, e) + cmul(e, e); // Mandelbrot (and Phoenix square part)
 }
@@ -183,6 +242,9 @@ fn fprime(z: vec2<f32>) -> vec2<f32> {
     } else if u.kind == KIND_LAMBDA {
         // Lambda: f'(z) = λ·(1-2z).
         return cmul(u.lambda_l, vec2<f32>(1.0 - 2.0 * z.x, -2.0 * z.y));
+    } else if u.kind == KIND_COMPLEX_MULTIBROT {
+        // f'(z) = p * z^(p-1).
+        return cmul(u.complex_power, cpow(z, u.complex_power - vec2<f32>(1.0, 0.0)));
     }
     return 2.0 * z;
 }

src/worker.rs

@@ -26,6 +26,8 @@ pub struct RefRequest {
     pub phoenix_p: (f64, f64),
     /// Distortion constant for the Lambda map (ignored by other kinds).
     pub lambda_l: (f64, f64),
+    /// Complex exponent for the Complex Multibrot kind (ignored by other kinds).
+    pub complex_power: (f64, f64),
 }
 
 pub struct RefResult {
@@ -107,6 +109,7 @@ fn compute(req: &RefRequest) -> Vec<[f32; 2]> {
             req.power,
             req.phoenix_p,
             req.lambda_l,
+            req.complex_power,
         )
     } else {
         compute_set_reference(
@@ -118,6 +121,7 @@ fn compute(req: &RefRequest) -> Vec<[f32; 2]> {
             req.power,
             req.phoenix_p,
             req.lambda_l,
+            req.complex_power,
         )
     }
 }