It is easy to set up a correspondence between the argument of a complex number and the hue of a color since both are measured as angles.  We use red to mean argument 0 (or 360) degrees;  green means 120 degrees, and blue means 240 degrees.  I have been using contour lines in saturation and value to represent the modulus of the complex value.  The contours distinguish moduli between 0 and 1, and between 1 and e, and e and e², etc.  Moduli between 0 and 1 (or any of these ranges) are essentially wrapped around a rectangle in saturation-value space with one corner at saturation = 1 and value = 1, and the opposite corner at saturation = 0.3 and value = 0.6. 

See Visual Basic code (.txt file).

The function f(z) = z  is plotted to the left below.  The contours appear as ridges with a shadow on the high modulus side.  Zeros always have a pinwheel of color, red to green to blue in a counterclockwise direction. For a simple zero there is one cycle of color; for a double zero two cycles, etc.  To the right below is a plot of a function with a simple zero, a double zero, and a triple pole.  Colors cycle in the reverse order around a pole, and there is always a set of modulus contours approaching the pole.