I've recently been learning about functions of complex numbers (to complex numbers), and I can't quite fit them into my head.
When I think about real functions, I tend to mentally visualize them as some sort of surface (a two dimensional surface, or a three-dimensional one). In this way, I can intuitively understand what it means for a function of 1, 2, or 3 real variables to be continuous or differentiable, and I can also visualize some problematic aspects of a function in order to get a hint about whether or not the function has these qualities.
This intuition breaks down when dealing with complex-valued functions, so I can't really make heads or tail of them.
Is there a good way of looking at this problem, from an intuitive perspective? Are there ways that you've successfully used in the past, such as in order to personally understand complex functions, or to teach students about them?
Since I'm also going to learn about functions of multiple complex variables, can these methods be extended to such functions?
To clarify, I am actually looking for such intuitive devices as Einstein's riding a beam of light.
Notes
This question is not about graphing/plotting/sketching complex functions, nor is it about visualizing functions in general, nor is it about visualizing complex numbers. There are many questions of this sort already, and I don't want this one to be perceived as a duplicate (unless someone can find an answer to this question that isn't about graphing, in which case it would be a great help).
I feel this is needed because of the large number of red herrings that use similar wording.
Best Answer
I tend to visualize functions from $\mathbb{C}$ to $\mathbb{C}$ as a vector field on $\mathbb{R}^2$. In other words, given an $f$, there is an arrow for every $x \in \mathbb{C}$ which starts at $x$ and ends at $x + f(x)$.
For $z \to e^z$, for example, if you look at a line with $\textrm{Re}(z) = u$, the arrows all have the same length $e^u$, but keep rotating uniformly as you move along the line.
To get a feeling for the global behaviour of a function $f$, it also often helps to look at the images of some family of simple geometric shapes.
Take $z \to z^2$, for example, and look at the images of the circles $M_r = \{z \,:\, |z| = r\}$. The image of $M_r$ is $M_{r^2}$, and each point on $M_r$ is mapped to a point with twice the angle on $M_{r^2}$.
For $z \to e^z$, you'd pick rectangles, not circles around the origin. Each rectangle is then mapped to a segment of some cone whose tip is the origin.