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    $\begingroup$ If the absolute values of all the complexes are the same, it should help to reformulate the problem as z1 = r Exp[I phi1] etc. and impose that r >= 0. $\endgroup$ Commented Apr 5, 2023 at 15:28
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    $\begingroup$ As I know it, the geometric interpretation of the solutions is the following: z1, z2, z3, z4 are vertices of a rectangle (maybe, a degenerate one) in the complex plane centered at the origin $\endgroup$ Commented Apr 5, 2023 at 17:21
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    $\begingroup$ @SjoerdSmit, you can go even further by taking r==1 $\endgroup$ Commented Apr 5, 2023 at 19:35
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    $\begingroup$ This is a 2-parameter family if we restrict length to be 1. $\endgroup$ Commented Apr 5, 2023 at 22:58
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    $\begingroup$ The geometric interpretation is more general: z1,z2,z3,z4 form a rhombus! $\endgroup$ Commented Apr 7, 2023 at 15:42