well i took that course from ronnie at brandeis, before he wrote the book, and as I recall the first edition had some mathematical flaws that were corrected in later ones. It seemed like a nice book to me, but I am not an expert on books on complex manifolds.
I also read kodaira and morrow, a little computational for my taste, and for one dimensional complex manifodls, i.e. riemann surfaces, there are many good books, like griffiths and harris, springer, gunning, but wells's book is beautifully printed, a plus, and carefully written.
At a little higher level I like hirzebruch's book on topological methods in algebraic geometry. other books I have on my shelf on on complex manifolds include varietes kahleriennes by andre weil, and the very terse pamphlet of chern - complex manifolds without potential theory, and also chern's notes from recife.
books on the local theory, i.e, several complex variables, include the classical books of hormander and gunning and rossi.
its a big subject. there are some nice local theorems, like extension theorems for holomorphic functions, then some ideas that are unique to more than one variable, like holomorphic convexity and domains of holomorphy, then there are basic differential operators and their regularity proeprties, like various laplacians, and the interplay ebetween harmonic and holomorphic functions. then there is the global theory combining the complex analysis with the topology, with concepts like chern classes, and sheaves, and vector bundles, and riemann roch theorems for counting the sections of holomorphic vector bundles in terms of topological invariants involving chern classes.
there are also generalizations to indices of elliptic operators by atiyah - singer. wells is probably a good place to start. or maybe gunning on compact riemann surfaces, for an intro to the ideas but in a simpler case of one dimension.
i.e. there are two special cases of complex manifolds that are often studied, namely the riemann surfaces, and the abelian varieties, or complex tori.
these are easier than the general cases and need less machinery. gunning has the nice idea to teach the general machinery but in the easier setting of the one dimensional case. abelian varieties are quotients C^n/Lattice, so can be studied globally on C^n, using quasi periodic functions.I am not really an expert here and if you master wells's book you should know more than me.
