Complex analysis is the good twin and real analysis the evil one: beautiful formulas and elegant theorems seem to blossom spontaneously in the complex domain, while toil and pathology rule the reals. ~ Charles Pugh
I was just wondering: What is the most pathological object/property you can think of in Complex Analysis?
EDIT: Also posted on MathOverflow as https://mathoverflow.net/questions/310418/pathology-in-complex-analysis