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I know that there exist many questions on this site on complex analysis books but my question is more specific than that. I am looking for recommendations for a concise complex analysis book but with a view towards algebraic geometry/ complex manifolds. I have studied smooth manifold theory before and it would be interesting to see it combined with complex analysis in the complex setting (all manifolds that I'd studied before were real manifolds). Any recommendations for such a book?

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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.

Georges Elencwajg
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    How does one pronounce your last name, @GeorgesElencwajg? – Soham Chowdhury Aug 19 '15 at 14:07
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    Dear @Soham: something like /'eləntsv'ajɡ/, but I'm happy with whatever pronunciation people want to use. If you know the sounds of French and German, just imagine that the great Austrian writer Stefan Zweig had a sister named Hélène Zweig... – Georges Elencwajg Aug 19 '15 at 22:05
  • Dear @Georges, that's really helpful. :) – Soham Chowdhury Aug 20 '15 at 17:14
  • Thanks, @Soham: I appreciate your sense of humour ! – Georges Elencwajg Aug 20 '15 at 17:17
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    No, I'm serious. Hélène Zweig is, like, the best pronunciation guide ever! Please tell me you prepared this years ago and use it whenever possible . . . – Soham Chowdhury Aug 20 '15 at 17:21
  • Dear @Soham, yes, I have been using that pronunciation trick for a long time, since I am a great admirer of Stefan Zweig . However the trick only works with erudite people like you who can pronounce both French and German. You are the more to be congratulated that, judging from your photograph and name, you might be Indian and have Hindi (or Tamil or Marathi or ...) as your mother tongue. – Georges Elencwajg Aug 20 '15 at 20:02
  • I am Indian, and my mother tongue is Bengali. Please don't call me erudite. :) I was lucky to be able to take advantage of optional German classes about three years back when I was in middle school. And French is just about everywhere. – Soham Chowdhury Aug 22 '15 at 04:02
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    Thank you for answering my rather indiscrete implicit questions, dear @Soham. – Georges Elencwajg Aug 22 '15 at 09:07
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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.

mrf
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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.

Philippe Malot
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There is Complex Geometry: An Introduction by Daniel Huybrechts. I can thoroughly recommend it when you come from the angle of algebraic geometry.

Jesko Hüttenhain
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