I'd like to get fractals out of these functions:

$\displaystyle f(z) = (1 + (-1)^z) \frac{z}{4} + (1 - (-1)^z) \frac{3 z + 1}{2} $

$\displaystyle g(z) = \frac{z}{4} (1 + cos[\pi z]) + \frac{3 z + 1}{16} (1 - cos[\pi z]) (3 - \sqrt{2} cos[(2 z - 1) \frac{\pi}{4}]) $

$\displaystyle h(z) = \frac{1}{4} (2 + 7 z - (2 + 5 z) cos[\pi z]) $

What I got now looks like this (for one of the equations), and though it can be used to generate some pretty pictures it isn't quite what I want (need). What I would like to see is something like this: Julia Set -- from Wolfram MathWorld

Code:

collatz = Compile[{{z, _Complex}},
Module[{x = z, i = 0, s = 10.^5, vic = 10.^5 + 0.0 I, maxit = 100},
While[i < maxit && Abs[x] < s, i++;
x = 1/4 (2 + 7 x - (2 + 5 x) Cos[\[Pi] x]);
If[Abs[x] < Abs[vic], vic = Abs[x], vic];]; vic]]
sol = Table[
collatz[x + I y], {x, 1.5, 2.5, .003}, {y, -0.5, 0.5, .003}] // Chop;
ListDensityPlot[sol, Mesh -> False, ColorFunction -> (Hue)]

Any help would be highly appreciated.