Although the virtues of complex numbers as things which can give you the square root -1 are mentioned early on in the educational process, very little seems to be done with them thereafter. If it weren't for complex impedances in electrical circuits, they might not even get much use at the college level.
Later on, quantum mechanics uses complex wave functions and mathematicians get to playing with Galois theory. But of course, one must certainly aspire to an algebraically complete field. Still, for the general public, the Mandelbrot set may provide the first actual contact with the intricacies of complex analysis, and then with a vengeance when it turns out that fixed points attract critical points all over the place, that intricate patterns repeat themselves over and over again, and yet there is a system to it all.
There are expositions of complex numbers at all levels, starting with Nahin's historical survey An Imaginary Tale [8] and including their application to geometry in Hahn's Complex Numbers and Geometry [7]. A good engineering exposition of complex numbers and their uses can be found in Guillemin's The Mathematics of Circuit Analysis [10]. Ahlfors' Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable [5] may well be the most authoritative presentations on an advanced level, whereas Knopp's volumes on Theory of Functions [6] are both comprehensive and quite classical.