It's obviously of more interest as a piece of outreach than as a piece of mathematics. Nevertheless I've always wondered about the e^ipi + 1 = 0 formulation. It seems ugly and ad hoc, and the connection between the "5 constants" is not all that meaningful.

That e^ipi = -1 is related to the much more profound observation that the complex numbers represent a sort of rotation into a previously unknown dimension of numbers.