So far, the motivations have been mostly of analytic flavor. Let me add an alien motivation.
Suppose you were an alien that only knows the complex number field $\mathbb{C}$ as a fundamental object (say, because it's Cauchy complete and algebraically closed) and considers $\mathbb{R}$ just as a minor object (much like we consider -say- the Eisenstein integers). And suppose you came up with the concept of increment and of derivative and then of differential: then, which Analysis would you study? And suppose you were able to imagine the notion of a manifold as a space with local coordinates in your favourite field: which geometry would you naturally study?
Of course, the answers are, respectively, "Complex Analysis" and "Holomorphic manifolds".
So, even if you're a Terrestrial, blinded by notions of linear order, isn't $\mathbb{C}$ a very fundamental object anyway? If so, then doing SCV and Complex Analytic Geometry is just doing the natural things with it.
