It depends on what one would count as "number theory". Even if one´s interest ultimately lies in "only" in algebraic number subfields of the reals, one could argue that there are objects attached to these structures (e.g. L- and Zeta-functions) that are number-theoretical and are of interest for themselves, that is, they are more than just a way prove other things. I think a nice example here is Artin´s conjecture on the holomorphy of the Artin´s L-function -> http://en.wikipedia.org/wiki/Artin_L-function. To elaborate a little bit on this, there is absolutely no reason why an arbitrary ordinary Dirichlet series should admit meromorphic continuation, let alone being entire.
Assuming that the purely algebraic and real-analytic approach turns out to suffice for almost all number-theorerical purposes, objects like the complex-valued L-functions may indeed not contribute so much more to our mathematical understanding, but they certainly do so to our "philosphical" (or if you want "meta-mathematical") understanding.
Moreover, in the end one of the things that matter are the interconnections between various fields of mathematics. In that case one a priori has to deal with structures even "higher" than the complex numbers, e.g. automorphic forms -> http://en.wikipedia.org/wiki/Langlands_program.