Complex analysis book for Algebraic Geometers

Solution 1:

An extremely good but shamefully underrated book is Łojasiewicz's Introduction to Complex Analytic Geometry.
The author, a renowned mathematician who discovered a very important inequality pertaining to analytic sets, managed to write a book with the contradictory qualities of giving very detailed explanations and proving quite advanced results.
Indeed the book starts with the definition of a ring (!) and goes on to prove a sophisticated version (due to Thom and Martinet) of the Weierstrass preparation theorem, the Puiseux theorem, Remmert's proper mapping theorem, Chevalley's theorem on images of constructible sets, the Cartan-Oka theorem on the coherence of the ideal sheaf of an analytic subset of $\mathbb C^n$, Chow's theorem on the algebraicity of analytic subsets of $\mathbb P^n$ (=the mother of all GAGA-type theorems!) , and much more.
The book is, alas, out of print but I suppose that many libraries have a copy.

A phonetic link
The undersigned happens to know that some patronyms are considered difficult to pronounce, so here is a link with the correct pronunciation of our author's name.

Solution 2:

Complex analytic and differential geometry by J.-P. Demailly is freely available and an excellent choice (especially for several complex variables) if you have a background in algebraic geometry.

Solution 3:

From Holomorphic Functions to Complex Manifolds by Klaus Fritzsche and Hans Grauert is a great introduction to complex manifolds with also a very nice course on several complex analysis. It's not really oriented on algebraic geometry, though. Hans Grauert is one of the founder of the complex geometry.