To give some background, I am mainly an analyst trained in harmonic/functional and do work on geometric pde's and spectral multipliers. Of late, I am trying to learn more about (research level) complex geometry. If I understand correctly, complex geometry means many things to many different people. To some, it is an extension of algebraic geometry, while many others would immediately think of the Newlander-Nirenberg or Yau's proof of Calabi conjecture.
I am looking for some kind of research “introduction'' to the analytic side of complex geometry. I want to develop a better appreciation of the pde theoretic tools that are relevant in complex geometry, the problems they can solve, and also some of the open problems in the field that are thought to be potentially amenable to analytic methods of attack. In other words, suppose an analyst wants to do research on complex geometry. What does he start by reading? Papers sometime down the line for sure, but initially they might be too specific/concentrated. I guess, if there were a textbook on complex geometry written by Yau, or Hormander, or Tao, amongst other people, that would be a starting point for me.
If my question is too broad/unfit for this site, I apologize. I realize that it is impossible that any single book/monograph/lecture note will cover all the analytic sides of complex geometry. But even a partial answer will be appreciated.
Lastly, just for example: if someone asked me what would be a good answer if someone asked the same question about real differential geometry? I would say I don't have a good answer, as the literature is just too huge. However, I would add that some of my favourites are Schoen/Yau's Lectures on Differential Geometry, Jost's Geometric Analysis, and perhaps Aubin's Nonlinear problems in Riemannian geometry. These books should certainly get one started.
Best Answer
1) There is a great book From Holomorphic Functions to Complex Manifolds by Fritzsche-Grauert.
It is very geometric and gives you the fundamentals on complex manifolds, including specialized topics, from Stein manifolds to compact Kähler manifolds, which in a sense are the two extremities in the spectrum of holomorphic geometry.
An important bonus is that Grauert was arguably the deepest complex analyst in the twentieth century.
It is very geometric but it will also warm your analyst's heart with sections on plurisubharmonic functions, Sobolev spaces and, Neumann operators,...
2) Another, even more analysis rich introduction, to complex geometry is Krantz's Function Theory of Several Complex Variables
There you will find harmonic analysis, regularity of $\bar \partial $ operator, $H^p$ functions and all sorts of integral representations.
3) Both books contain introductions to the indispensable tool of sheaves and their cohomology, which actually had their first applications in complex analysis, and were foreshadowed in Oka's groundbreaking solution to the Levi problem.
There are other good books, by Fuks, Griffiths, Hörmander (mentioned in abx's comment), Huybrechts, Ohsawa, Range, Wells,... but for me the most comprehensive introductions are 1) and 2).
4) Welcome to that enchanting land of complex geometry (full disclosure: that was where I started doing research!) and good luck!