Complex analysis is the study of functions of a complex variable, i.e., functions $f\left(z\right)$ where $z$ and $f\left(z\right)$ lie in $C$. Complex analysis is the good twin and real analysis the evil one: beautiful formulas and elegant theorems seem to blossom spontaneously in the complex domain, while toil and pathology rule in the reals. Nevertheless, complex analysis relies more on real analysis than the other way around.

--Pugh, Real Mathematical Analysis (p. 28)