Vectorfields
One could use the usual visualization of vectorfields, fields of forces to plot complex functions. The idea is to place some little arrows with orientation according to the argument and size proportional to the magnitude of the function value at grid points of a domain.
On the figure below one can see a polynomial of degree two:
\(f(z) = z^2-1\)
The two zeros are at -1 and 1. Arrows of length zero 'are to be found' here.
The exponential function is worth to look at this way too.
\(f(z) = \exp\,z\)
One can observe the periodicity and the typical behaviour of absolute values and angles.