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)].

Seriously, 25 bytes

,i││-++½)+-+½)++-½)±+++½)

Takes input as a list: [x1,y1,x2,y2]

Same strategy as my Python answer, but in Seriously!

Explanation:

,      get input
i      flatten list
││     duplicate stack twice, so that we have 4 copies of the input total
-++½)  calculate the first x-value using the formula (x1-y1+x2+y2)/2, and shove it to the end of the stack
+-+½)  calculate the first y-value using (x1+y1-x2+y2)/2, and shove it to the end of the stack
++-½)  calculate the second x-value using (x1+y2+x2-y2)/2, and shove it to the end of the stack
±+++½) calculate the second y-value using (-x1+y1+x2+y2)/2, and shove it to the end of the stack

Try it online

TI-BASIC, 16 bytes

For a TI-83+ or 84+ series calculator.

Input X
i∟X+.5sum(∟X-i∟X

Unless I misunderstood, OP said they were fine with taking input and output as complex numbers. The i here is the imaginary unit, not the statistics variable.

TI-BASIC has a mean( function, but annoyingly it doesn't work with complex lists, throwing an ERR:DATA TYPE.

Input in the form {1-i,4+2i} for [[1,-1],[4,2]]. Output is in the form {4-i 1+2i} for [[1,2][4,-1]].