z_n+1 = z_n²+z_c, z_c = -0.63-0.407i
This is a typical Julia Set fractal. Julia Sets were discovered by Gaston Maurice Julia.

Julia Set Fractal – Mathematica 4.2: 4/15/04, POV-Ray 3.6.1: 7/4/06

(* runtime: 36 seconds *)
Julia[zo_] := Module[{z = zo, i = 0}, While[i < 100 && Abs[z] < 2, z = z^2 + zc; i++]; i];
zc = -0.63 - 0.407 I;
DensityPlot[Sqrt[Julia[x + I y]], {x, -1.5, 1.5}, {y, -1.5, 1.5}, PlotPoints -> 275, Mesh -> False, Frame -> False, ColorFunction -> Hue];

POV-Ray has a built-in function for these fractals:
// runtime: 0.5 second
camera{orthographic location <0,0,-1.5> look_at 0 angle 90}
plane{z,0 pigment{julia <-0.63,-0.407>,200} finish{ambient 1}}