Peterzil and Starchenko began this by developing the basics of complex analysis (Cauchy’s theorem, Taylor series, residues…) within an arbitrary o-minimal expansion of a real closed field. I look at more advanced topics from such a definable viewpoint (eg the Riemann Mapping Theorem) although to make any progress I have to restrict myself to (o-minimal) expansions of the real field itself. I am, of course, motivated by Zilber’s quasiminimality conjecture.