I consider it kind of important that the notation for expressions like “A²” doesn’t depend on whether A is an integer, real number, complex number, matrix, rand

I consider it kind of important that the notation for expressions like “A²” doesn’t depend on whether A is an integer, real number, complex number, matrix, random variable, etc., (even if the results do) or what the specific domain is, but if you feel like it’s important to embed all of that context in the exponent operator... give it a try :) (And whether “2” is integer, real, rational, complex, etc)