TITLE: Pathology in Complex Analysis
QUESTION [35 upvotes]: 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.

REPLY [5 votes]: Another complex dynamics example:
Suppose $0 < \lambda < \frac{1}{e}$. The Julia set of $\lambda e^z$ can be divided into a set $E$ of "endpoints" and a collection of "hairs" connecting these endpoints to $\infty$. Mayer proved in 1990 that $E$ is totally separated, but $E \cup \{\infty\}$ is connected.