((a+ib)/(a-ib))2=(a+ib)4/(a2+b2)2;
((a-ib)/(a+ib))2=(a-ib)4/(a2+b2)2;
add these together:
((a+ib)4+(a-ib)4)/(a2+b2)2.
(a+ib)4=a4+4a3ib-6a2b2-4aib3+b4;
(a-ib)4=a4-4a3ib-6a2b2+4aib3+b4;
add these together:
2a4-12a2b2+2b4;
divide by (a2+b2)2:
2(a4-6a2b2+b4)/(a2+b2)2, which has only real components, making its conjugate the same as itself.