I have 3 general abstract reasons to care about complex analysis in a single variable:


1. The laplacian is, up to a constant multiple, the only isometry invariant PDO in the plane, and so it is abstractly very important. Holomorphic functions are intimately related to harmonic functions, so holomorphic functions are important.
2. Holomorphic mappings are conformal if they have nonvanishing derivatives
3. Many results which are real variable in nature are most easily understood in the light of complex analysis (factorization of real polynomials, radius of convergence of real power series, etc)


I currently have no such justification for several complex variables. The connection to harmonic functions mostly breaks down. Conformality breaks down. I have not seen applications to real variable phenomena. 


Can anyone give me some insight into the big picture here? Where do the phenomena studied in several complex variables (domains of holomorphy and their ilk) come up in the rest of mathematics?



Fundamental motivation for several complex variables
On my opinion, there are two very general reasons why analytic functions are important.


1. Solutions of many (almost all) important differential (and functional) equations are analytic. 
For example, 
all elementary and special functions arise in this way. Moreover, they are also usually 
analytic functions of parameters.
2. Harmonic analysis. Fourier (Laplace) transforms. 
This includes generating functions as a special case.
They are in many cases analytic.


I think almost every occurence of analytic functions in mathematics and in real life can be traced 
to one of these two general sources.


Both these fundamental motivations apply not only to analytic functions of one variable but to
analytic functions of several variables.


Historically, it seems that the first analytic functions of several variables that were studied
were Abelian functions, which occur in the inversion problem of Abelian integrals
(that is they occur as solutions of certain systems of differential equations). These differential
equations appear both in pure mathematics (inversion of Abelian integrals) and in physics
(integrable systems). They were studied long before the general theory of several complex
variables was developed.


Then comes the second source (already in XX century): Wiener-Paley theorem and its multi-dimensional
generalizations, Fourier analysis, which itself was developed as a method of solving PDE.


You specifically mention applications in algebraic geometry? Just open Griffiths Harris book!
And for applications in "real analysis", look at least at the table of contents of Hormander's
Linear differential operators!


