I think the point is to look at it the other way: we invented/chose i so that exponentiation (and all other functions) maintain the same properties for the complex numbers as they have for the reals. The whole goal of defining the complex numbers is to extend R in a way that maintains all of the properties of functions on R.
I dunno if Im just not coming across clearly or my thinking is too abstract, but when you say
"how exponentiation works"
you are referring to how exponentiation works for real exponents
You would have to prove that it works the same way for i exponents before you can go further.