general topology - Open set in $mathbb{C}$ with non-trivial boundary

When studying complex analysis - or even real analysis for that matter - we most times consider open sets $\Omega \subset \mathbb{C}$ (or $\Omega \subset \mathbb{R}^2$) having smooth curves as its boundaries (Ex.: an open disk), or at least smooth by parts (Ex.: a triangle).

I assume that is necessary so we can deal with line integrals, Cauchy Formulas and that kind of thing. But from a purely topological curiosity, what are the possibilities for the boundary of an arbitrary open set? How complicated can it get? Can we give non-trivial examples explicitly?

Thanks!