Actually, even the definition of this form requires i, because it's
e^(i)(pi/2) = i.
The simple assumption that sqrt(-1) = i is problematic because you can get something like
i^2 = (sqrt(-1))^2 = sqrt((-1)^2) = 1.
The wonder and surprise of complex numbers is that assuming seemingly arbitrary properties of a constant like i lead to a number of deep and beautiful results.
sqrt(-1) = sqrt(e^Pi) = e^(Pi/2) = i
So the new concept was really a 2D plane for numbers, not a new definition for sqrt of negative integers.