For a complex number z, let Re $\left(z\right)$ denote the real part of z. Let S be the set of all complex numbers z satisfying $z^4-{\left|z\right|}^4=4i{z}^2,$ where $i=\sqrt{-1}.$ Then the minimum possible value of ${|z_1-z_2|}^2,$ where $z_1,z_2\in S$ with $\text{Re}\left({\text{z}}_{\text{1}}\right)>0$ and $\text{Re}\left({\text{z}}_2\right)<0,$ is _________

$$\begin{gathered} Let z=x+iy \\ {z}^4-{\left|z\right|}^4=4i{z}^2 \\ \Rightarrow z^4-z^2{\overline{z}}^2=4i{z}^2 \\ \Rightarrow z=0 or z^2{\overline{z}}^2=4i \end{gathered}$$

$$\begin{gathered} \Rightarrow z=0 or 4xyi=4i \\ \Rightarrow z=0or xy=1 \\ \Rightarrow xy=1 \end{gathered}$$

So, locus of z is rectangular hyperbola.

$\therefore {|z_1-z_2|}_{\min }^2=8$