Question

How long did it take you to be able to read and get information off of complex graphs with colour?

Answer 1

I *never* did that. It is not a normal way for anyone to think about complex analysis.

It can be useful to consider two planes side by side. But that color stuff? (Excuse me, colour stuff?) Forget it. You are wasting your time if you think it is important.

Answer 2

Graphing complex functions this way is a cute trick, but as you're finding, it is pretty difficult to get information from and I don't think most people ever rely on it in any serious way. I never bothered to use it at all even though I am generally a pretty visual person, and I got through my qualifying exam in complex analysis just fine.

You can think about the real part and the imaginary part as two real-valued functions on the plane, and that can be visualized in a straightforward fashion. You can also think of complex functions as transformations on the plane (eg, z^2 doubles angles, e^z maps a vertical strip to an annulus and a horizontal strip to a pie slice).

Answer 3

It's a skill like any others. I still catch myself staring vacantly at complicated 2d graphs trying to decipher them sometimes, and 3d models had a whole extra dimension. You'll get there if you keep at it, but it's a non-trivial increase in complexity.

I'd say it took doing a research project in which I had to regularly visualize 3d graphs (I could have plotted them, but it was quicker to just visualize them) for a year before I was really comfortable with them.