The Wolfram Language transparently works with complex variables throughout, not only numerically, but also symbolically—often relying on original results to handle intricate branch cut and other issues.
ParametricPlot — plot conformal mappings
Integrate — symbolic integrals taking account of complex variables
PrincipalValue — option for specifying whether to take principal values
NIntegrate — numerical integration around contours in the complex plane
Residue — residue at a pole
ComplexExpand — symbolically expand into real and imaginary parts
Reduce — reduce polynomial systems over complex variables