(*Adapted by Rafael Luís: rafaelluis@netmadeira.com
updated to Mathematica v.10 by Ross Moore *)

theFunction[z_] := z^3-1;
iterateThis[z_] = z - (theFunction[z] - 1)/theFunction'[z];
ContourPlot[
  Arg[
      FixedPoint[ iterateThis, x + I y, 30 ]],
        {x, -1.5, 1.5}, {y, -1.5, 1.5},
        ContourLines -> False, PlotPoints -> 175,
        ColorFunction -> Hue, Axes -> False, Frame -> False ]