I'm surprised that more emphasis hasn't been given to the key point
One cannot over-emphasize that passing to complex numbers often permits a great simplification by linearizing what would otherwise be more complex nonlinear phenomena. One example familiar to any calculus student is the fact that integration of rational functions is much simpler over C $\mathbb C$ (vs Rvs. $\mathbb R$) since over C partial fraction decompositions involve at most linear (vs quadratic) polynomials in the denominator. Similarly one reduces higher-order constant coefficient differential and difference equations to linear (first-order) equations by factoring the linear operators over $\mathbb C$. More generally one might argue that such simplification by linearization was at the heart of the development of abstract algebra. Namely, Dedekind, by abstracting out the essential linear structures (ideals and modules) in number theory, greatly simplified the prior nonlinear theory based on quadratic forms. This permitted enabled him to exploit to the hilt the power of linear algebra(both conceptually, and computationally). Examples abound of the revolutionary power that this brought to number theory and algebra - e.g. For a pretty for one little-known example gem see my recent poston conductor ideals explaining how Dedekind's notion of conductor ideal beautifully encapsulates the essence of elementary irrationality proofs - in this recent thread on Dedekind's methodology http://mathoverflow.net/questions/30220/abstract-thought-vs-calculation/30313#30313of n'th roots.
