The reason I care about functions with several complex variables is the resolvent formalism. To solve a problem in linear algebra, you translate it into a problem in complex analysis (with several variables) and allow tools like Cauchy's Theorem and the Argument Principle (for functions of a single complex variable) to chew it up. The nicest proof I know that a matrix has a Jordan normal form proceeds in this way, and is nicely explained HERE. I presented this proof in the introductory Complex Analysis course I have just finished teaching as an illustration of the power of complex analysis, never mind the number of variables!
The nicest thing about resolvents is that they work just as well when the dimension is infinite (studying operators on Hilbert spaces instead of plain old linear transformations on finite dimensional vector spaces). They are thus a basis for Fredholm Theory.
