I'm confused. How is this simpler? Is there something in (-1)^(2x) that can easily understood by staring at the complex plane? It seems mostly that you've gotten rid of "e", but one of the goals of Euler's formula IMO is to explain what "e^(i …)" means so I'm not sure how this variant is useful.

Sorry but this is pretty bogus. (-1)^x is only well defined when x is an integer. This is generally the case for r^x whenever r isn't a positive real number. For example, when x = 0.5, r has two distinct square roots. Sure, you can choose one of them arbitrarily and declare it to be the value of r^0.5 (and math libraries typically do this), but there's unfortunately no good way to make this arbitrary choice consistently for all values of r simultaneously.

Now define exponentiation by a non-integer.