What's the difference between $|z|^2$ and $z^2$, where $z$ is a complex number? I know that $|z|^2=zz^*$ but what is $z^2$? Is it simply $z^2=(a+ib)^2$?

$z^2 -|z|^2 = a^2 + 2abi - b^2 - a^2 - b^2 = 2abi - 2b^2 = 2b(ai-b) = 2bzi$
Similarly, $z^2+|z|^2=a^2+2abi-b^2+a^2+b^2=2a^2+2abi=2a(a+bi) = 2az$
