I have 3 general abstract reasons to care about complex analysis in a single variable:
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.
Holomorphic mappings are conformal if they have nonvanishing derivatives
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?