When people say that the square root of -1 is i they are only partly correct. Just as 4 has the two square roots 2 and -2, -1 has the two square roots i and -i. Both these numbers give -1 when squared and are therefore square roots of -1. The difference between any complex number (a+bi) and its complex conjugate (a-bi) is therefore just that we use the alternate square root of -1.