Starting with just the property $E(x + y) = E(x)E(y)$, one can prove quite a lot of the main properties of the exponential function on real numbers. For example, $E(0) = 1$, and $E'(X) = E(X)$, and $E(nx) = E(x)^n$ all follow from a straight-forward application of definitions.

To move this in to complex analysis and Euler's formula, it seems to me that the key property that needs to be proved is $E(\overline{z})$ = $\overline{E(z)}$.

Is there a nice way to prove this that is in the spirit of this particular line of reasoning?

Edit: For example, Chapter 8 of Baby Rudin follows this reasoning, but resorts to the definition $ E(x) = \Sigma_{n\ge0}(\frac{z^n}{n!})$ to prove the conjugation property.