Another nice way one could preserve the complex data when visualizing it would be to make a 3d color mesh and display the imaginary components as the height in z and the real component as the color scale (or vice-versa).
Edit* now I am trying to think if there would be a clever way to show the abs, Re and Im values in one 3d plot, but drawing a blank. Maybe tie Im to the alpha value to make the transparency change as the imaginary component goes up and down? It would just require mapping the set of all numbers from -inf:inf to 0:1, which is doable in a 1-1 transformation iirc since they both have cardinality C. I think it would be
alpha = 1 - 1/(1-e^{Im(z)})
Which looks a lot like the equation for Bose-Einstein statistics in Stat. Mech. I was never very good at complex analysis or group theory though, so I don't really know what to make of that.