Imaginary numbers become useful when working with a lot of advanced mathematics, and make equations work nicely.

A simple example though, is square rooting a negative number as you mentioned, but there's a little more to it. square_root(-1) = i, but that doesn't mean "no solution possible," it means that it's not possible at that moment. For example, if I plug that result into the function "f(x)=2+x^(2)" it now becomes a valid answer, because I am squaring the "impossible," answer of square_root(-1), cancelling out the square root