Square side pinpointing
Seriously, 11 bytes
A port of my TI-BASIC answer. Calculates mean(X)+i*(X-mean(X)).
,;Σ½;)±+ï*+

Explanation:
,           Read input
;           Duplicate
Σ½          Half the sum (the mean) of the top copy
;           Copy the mean
)           Rotate stack to the left
            Now there's a copy of the mean on the bottom
±+          Negate mean and add to input list
ï*          Multiply by i
+           Add to mean

Input as a list of two complex numbers: [1-1j,4+2j], and output in the same format: [(4-1j), (1+2j)].