For a complex number z , let Re (z) denote the real part of z . Let...

${z}^{4} - | z{|}^{4} = 4i{z}^{2} ⇒ {z}^{4} - {\left(z\overset{¯}{z}\right)}^{2} = 4i{z}^{2}$

$⇒ {z}^{2}\left({z}^{2} - \overset{¯}{z}2\right) = 4i{z}^{2} ⇒ {z}^{2} - \overset{¯}{z}2 = 4i$

$⇒ \left(z + \overset{¯}{z}\right)\left(z - \overset{¯}{z}\right) = 4i$

$⇒ \left(\dfrac{z + \overset{¯}{z}}{2}\right)\left(\dfrac{z - \overset{¯}{z}}{2i}\right) = 1 ⇒ xy = 1$

for ${z}_{1} \text{ and } {z}_{2} ⇒ {x}_{1}{y}_{1} = 1$ and ${x}_{2}{y}_{2} = 1$

${x}_{1} \text{ and } {x}_{2}$ are of opposite sign, similarly

${y}_{1}$ & ${y}_{2}$ are of opposite sign

$⇒ {x}_{1} > 0, {y}_{1} > 0, {x}_{2} < 0, {y}_{2} < 0$

Now ${\left|{z}_{1} - {z}_{2}\right|}^{2} = {\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}_{2}\right)}^{2}$

$= {x}_{1}^{2} + {x}_{2}^{2} + {y}_{1}^{2} + {y}_{2}^{2} - 2{x}_{1}{x}_{2} - 2{y}_{1}{y}_{2}$

$= {x}_{1}^{2} + {x}_{2}^{2} + {y}_{1}^{2} + {y}_{2}^{2} + {x}_{1}\left(- {x}_{2}\right) + {x}_{1}\left(- {x}_{2}\right) + {y}_{1}\left(- {y}_{2}\right) + {y}_{1}\left(- {y}_{2}\right)$

$\ge 8{\left({x}_{1}^{2} \cdot {x}_{2}^{2} \cdot {y}_{1}^{2} \cdot {y}_{2}^{2} \cdot {x}_{1}\left(- {x}_{2}\right) \cdot {x}_{1}\left(- {x}_{2}\right) \cdot {y}_{1}\left(- {y}_{2}\right) \cdot {y}_{1}\left(- {y}_{2}\right)\right)}^{1 / 8}$

$\ge 8{\left({\left({x}_{1}{y}_{1}\right)}^{4}{\left({x}_{2}{y}_{2}\right)}^{4}\right)}^{1 / 8}$

$\ge 8$.