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).