As it turns out, the Julia set of $f(z) = z^2 + c$ is either "in on piece" or "totally dusty". This leads to the definition of a Mandelbrot set $M$, which is the set of all parameters $c \in \mathbb{C}$ for which such Julia set is connected:

$$\begin{aligned} M = \{ c \in \mathbb{C}: J(z^2 + c) \text{ is connected} \} \end{aligned}$$

Here are two amazing theorems:

$J(z^2 + c)$ is connected if and only if 0 does not belong to $A(\infty)$ !

In other words, $\{f^n(0)\}$ remains bounded under iteration. Furthermore,

A complex number $c$ belongs to $M$ if and only if $\lvert f^n(0) \rvert \le 2$ for all $n \ge 1$ !

where $f(z) = z^2 + c$.

You can find more info at Analysis of a Complex Kind.