I don't have any knowledge on transcendence proofs. I just heard that Lindemann proved that for any $\alpha \in \mathbb R^\*$ algebraic, $e^\alpha$ is transcendental. Then, since $i$ is algebraic, and so is $e^{i\pi}=-1$, $\pi$ must be transcendental, because otherwise $e^{i\pi}$ would be transcendental.

My question is the following. Of course this proof is "valid", I'm not questioning that. But, since the complex exponential function is constructed as an extension of the real exponential by means of the power series (as far as I know, although maybe it is possible to define it in another way), and although its uniqueness can be proved, does this proof of the transcendence of $\pi$ have as much force as a proof which would not use the complex exponential function ? (As, I think, is the case for the transcendence of the Liouville number and $e$).

To put it in another way : when we talk about $e^{i\pi}$ or $3e^{i\frac{{\pi}}{5}}$, is the $e$ used in the same way as in $e^5$ ; I mean, are we really talking about the number $e$ ? Or is it just a "manner of talk", something which simplifies calculations and helps us find Euler's formulas when we have forgotten them ? If the definition of $e^{i\theta}$ is $cos\space\theta+i\space\sin\space \theta$, then it is something different from "the real number e that we see in real analysis", and then how can we be sure that the property I mentioned, ie the fact that $e^\alpha$ is transcendental if $\alpha$ is algebraic, is still valid for complex values ?

I hope my question is clear enough.