Yes, that would be ListDensityPlot
.
Some example code:
SqrtFractalSetup[{xmin_, xmax_}, {ymin_, ymax_}, n_] :=
With[{xminn = N[xmin], yminn = N[ymin],
xstep = N[(xmax - xmin)/n], ystep = N[(ymax - ymin)/n]},
re0 = Table[r, {r, xminn, xmax, xstep}, {i, yminn, ymax, ystep}];
im0 = Table[i, {r, xminn, xmax, xstep}, {i, yminn, ymax, ystep}]];
SqrtFractalDraw[c_, {xmin_, xmax_}, {ymin_, ymax_}, steps_] :=
Module[{quadrant}, {re, im} = {re0, im0};
Do[{re, im} = {(c - Sqrt[Abs[re]] + Sqrt[Abs[im]])^2,
(1 - Sqrt[Abs[im]] - Sqrt[Abs[re]])^2}; , {steps}];
quadrant = Abs[re + I*im];
ListDensityPlot[quadrant, Mesh -> False, ColorFunction -> Hue, PlotRange -> All]];
SqrtFractalSetup[{-1, 1}, {-1, 1}, 400];
Animate[SqrtFractalDraw[
N[0.6 + 0.6*Cos[c] + (0.25 + 0.25*Sin[c])*I], {-1, 1}, {-1, 1}, 7],
{c, (2*Pi)/9, 2*Pi - (2*Pi)/9, (2*Pi)/36}]
and here is a related video from WWDC 2003: