Question

For a complex number $z$, let $Re(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^{4}-|z|^{4}=4 i z^{2}$, where $i=\sqrt{-1}$. Then the minimum possible value of $\left|z_{1}-z_{2}\right|^{2}$, where $z_{1}, z_{2} \in S$ with Re\left( z _{1}\right)$$>0 and $Re\left( z _{2}\right)$ $<$ 0, is ______

Answer 1

$|z_1 – z_2|^ 2 = 8$