An "elementary" approach to complex exponents? Is there any way to extend the elementary definition of powers to the case of complex numbers? 
By "elementary" I am referring to the definition based on $$a^n=\underbrace{a\cdot a\cdots a}_{n\;\text{factors}}.$$ (Meaning I am not interested in the power series or "compound interest" definitions.) This is extended to negative numbers, fractions, and finally irrationals by letting $$a^r=\lim_{n\to\infty} a^{r_n}$$ where $r_n$ is rational and approaches $r$.
For a concrete example, how would we interpret $e^i$ in terms of these ideas?

So here's a good place to start
$$e^{i\theta}$$
Is interpreted as the complex number that is formed if you form a circle of radius 1 in the complex number field. And starting from the point 1 + 0i you move along the circle for an angle $\theta$ to a new number in the complex number field:
$$\sin(\theta) + \cos(\theta)i)$$
Now note that ANY complex number is of the form
$$r e^{i\theta}$$
Where $r$ is the absolute value of the complex number (or) it's distance from the point 0.
The value of the complex exponential simply indicates the angle. In other words we have polar coordinates here.
Then taking exponents becomes quite obvious with it simply distributing over both items which can be factored into complex exponential themselves.
If 'factor' your exponentials into a product of numbers of this format then it becomes intuitive what physically is occurring.
Hope that helps :)
