To me elementary number theory encompasses those results that can be completely derived starting with the definition of prime and composite within a book of reasonable size (say, a few hundred pages at most) without using any material not found in the first three years of a traditional undergraduate mathematics education (so the level of such a book would be appropriate for the typical junior or senior mathematics major). Therefore, I would not immediately exclude complex numbers (as Arminius mentioned there are some beautiful proofs of quadratic reciprocity using complex numbers), but I probably would exclude very heavy-duty complex analysis that one often finds in proofs of, say, the prime number theorem.

An interesting point is that there are several approaches one can take to developing number theory starting with the very basic definitions. One can take a predominantly algebraic approach, a mostly analytic approach, a computational approach, or one can try to mix these together in some way. The result is that elementary number theory really encompasses several books, each starting from the basic definitions but developing the subject from a different perspective.