Hopalong Fractal

a = -12;
b = +0.3;
c = -0.56;
f[{x_, y_}] := {y - Sign[x]*Sqrt[Abs[b*x - c]], a - x};
pts = NestList[f, {0, 0}, 100000];
Graphics[{
  PointSize[Small], Opacity[0.3],
  Table[{
    ColorData["DarkRainbow"][i/101],
    Point[
     pts[[Floor[i * Length[pts]/101] ;;
       Floor[(i + 1) * Length[pts]/101]]]]
    },
   {i, 100}]
  }]