Pathology in Complex Analysis 
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.
 A: 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.
A: I would say that Mandelbrot set (like similar fractal objects coming from complex dynamics) can be seen as a "pathological" object, at least from the point of view of regularity (the boundary is nowhere differentiable, for instance). 
A: I don't know how you want to define "pathological", but some of the corollaries of Runge's theorem give you functions  with interesting properties.  One of mine: there is a rational function $f$ such that for every holomorphic function $g$ on the open unit disk $\mathbb D$, $g$ or $g-f$ has a zero in $\mathbb D$.
This is American Mathematical Monthly problem 6520, solution at www.jstor.org/stable/2323638
A: A natural boundary is probably rather pathological, in the same spirit as that continuous-everywhere-but-differentiable-nowhere and smooth-everywhere-but-analytic-nowhere functions are in real analysis. In particular, it consists of a function which has a property (analytic) that one might intuitively expect would lead to another (analytic continuation, to at least some extent), but doesn't.
A simple example is the series function:
$$f(z) = \sum_{n=0}^{\infty} z^{2^n}$$
This function is defined on $|z| < 1$, but the circle $|z| = 1$ is singular, and the series thus both converges on the maximal domain and forbids any extension beyond it. The latter curve is thus a natural boundary - an enclosing wall singularity that prevents any further extension of the function's domain to an area of nontrivial measure.
A: In an old MO question of mine, I had wondered the following (I'm quoting my question):

Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halves $D_1$ and $D_2$.
Let f : D→ℂ be a continuous function that is holomorphic on the interiors of $D_1$ and $D_2$. Is f then necessarily holomorphic?

The answer turns out to be no.
A: I'd say that multivariable complex analysis is more complicated relative to single variable complex analysis than multivariable real analysis is to single variable real analysis. There are new phenomena that could charitably be called 'rich' and uncharitably be called 'pathological.'
For instance, it's well known that there are no non-constant holomorphic functions on 1D compact complex manifolds, but there are always non-constant meromorphic functions. In higher dimensions there are compact complex manifolds without even any non-constant meromorphic functions. 
Another thing is related to The_Sympathizer's answer: Any open set in $\mathbb{C}$ can be the 'domain of holomorphicity' of a holomorphic function, i.e. a domain which beyond which the function cannot be analytically extended. In higher dimensions this is no longer true and characterizing the open sets which are domains of holomorphicity becomes somewhat complicated.
A: The rigidity of complex domains in higher dimension For example the unit ball in $\mathbb{C}^2$ is not  holomorphic equivalent to the unit cube.
