It is true that sqrt(1) = 1 if you are using the principle square root function, but you only specify that later, not in the "proof". I was merely trying to point out that you were wrong when you said that for a=a to imply that sqrt(a)=sqrt(a) we need to take the principle square root. There are two possible roots, one where we do have that sqrt(x^y) = x^(y/2)(ie. sqrt(i^4) = i^2)and one where sqrt(1) = 1.
I am not trying to say that you were wrong, obviously the problem is with the sqrt step. It's just that the way you put it seemed misleading, at least to me. What I was trying to say is that it is not necessary to assume that sqrt is the principle square root function. I probably should have written something like "There is actually nothing wrong with taking the general square root of a complex number".
Also, I never said that i^4 is imaginary