Identities

The usual algebraic identities still work on complex numbers — $i$ just behaves like any other symbol, as long as you fold $i^2$ back into $-1$ wherever it shows up.

$(a+bi)^2 = a^2 - b^2 + 2abi$

$(a+bi)(a-bi) = a^2 + b^2$

That second one is the difference-of-squares identity — except since $(bi)^2 = -b^2$, it turns into a sum of squares instead.

Example: $(2+3i)^2 = 4 - 9 + 12i = -5 + 12i$.