Of fractals
There are plenty of fractals that are defined on the complex plane such as the Mandelbrot set and the Julia sets.
These are tipically not thought of as complex functions, still they are closely related: the methods used to visualize them also give a coloring of the complex plane. Thus they should be mentioned.
The Mandelbrot set.
Fractals like this can be defined using properties of an iteration. The color at a given point is determined by the characteristics of the iteration initialized at that point. (E.g. black if it is bounded, white if it's not. One may also monitor how many steps are required to exceed a given bound. Many different colorings may be thought of.)