I say this with trepidation as these things can be very polarizing, but publish and be damned I will:
how about that $$(\mathrm{cos}\,\theta + i\, \mathrm{sin} \,\theta)^n = (\mathrm{cos}\,n\theta + i\, \mathrm{sin} \,n\theta)=e^{in\theta}$$ and while we're at it, that $i^i = e^{-\pi/2}$ (after posting this I see that one of the comments has already mentioned $e^{i\pi}+1=0$). In the spirit of one of the comments to OP, rather trivial thanks to Argand etc, but still absolutely captivating. When you learn these things as a kid the simplicity of the proof given the perspective you have no doubt been given belies their import.
If too hackneyed an example please accept my apologies!
(NB: as correctly pointed out, we should define what we mean by $z^\alpha$ in general, namely $\mathrm{exp}(\alpha \log z)$ where $\mathrm{log}\, z =\mathrm{log}\,|z| + i \, \mathrm{arg} \,z$, hence we admit a multiplicity of values unless we restrict $\mathrm{arg} \,z$, which I'll do here to $0\leqslant\mathrm{arg} \,z\lt 2\pi$, so that $\mathrm{arg}\,i:=\pi/2$ )