Complex Analysis
A few examples of what you can ask Wolfram|Alpha about:
do basic arithmetic on complex numbers
apply functions to complex numbers
Complex Functions
compute properties of a function of a complex variable (use the variable z)
plot a complex function
compute the residue of a function at a point
compute residues at the poles of a function
compute residues at poles in a specified domain
Riemann Surfaces
visualize a Riemann surface