Question bank/JEE Advanced/Mathematics

For a complex number $z$, let $\operatorname{Re}(z)$ 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 what is the minimum possible value of $\left|z_{1}-z_{2}\right|^{2}$, where $z_{1}, z_{2} \in S$ with $\operatorname{Re}\left(z_{1}\right)>0$ and $\operatorname{Re}\left(z_{2}\right)<0$?

Source: JEEBench (IIT Delhi) / JEE Adv 2020 Paper 2