That is not actually the way to define e^x for complex numbers. There are a few ways you can do so. The most natural way is as the unique solution to the differential equation

df/dx = f

The solutions are e^x + const; requiring f(0) = 1 gives you the choice const = 0.

Another way to define it is as the infinite series

e^x = \sum_{i=0}^inf x^i/i!

It is pretty easy to show this definition is equivalent to the previous one. A third way is as

lim_{n -> inf} (1 + x/n)^n

This definition comes from the intuition that e^x represents the limit of continually compounding interest. As before, it is pretty easy to show that it is equivalent to the previous two.

In any case, all these definitions extend directly to complex values of x. The fact that

e^{ix} = cos(x) + i sin(x)

holds (and hence Euler's identity holds) is a consequence of a more natural definition, not the typical definition.