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

People often like to talk about elegant "miracles" in Complex Analysis. However, what's are "pathological" objects/properties in Complex Analysis?

EDIT (09/13/18): Also posted as https://math.stackexchange.com/questions/2912320/most-pathological-object-in-complex-analysis

EDIT: Changed the wording of the question.