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So complex numbers solve all polynomials, appear as eigenvalues, appear in intermediate calculations in solving cubics, relate trig to hyperbolic functions, can be used to contour integrate real functions more easily, can represent fourier series more compactly, describe calculations about wave phenomena, used in potential theory and conformal maps.

But what are some unusual, not well known or just advanced applications of complex numbers that one would be unlikely to encounter in an undergraduate mathematics degree ?

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    You could flick through this [book on special functions](http://books.google.ch/books?id=po-6Yxz851MC&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false) to see some applications of contour integration to prove certain identities and other useful stuff like asymptotic estimates.2012-07-13

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