I am looking at self similar sets in $\mathbb{C}$ defined as the fixed set or a sequence of contractions or an iterated function system. I am currently trying to classify these sets by how they are connected. I have come up with 3 classifications so far and think that all self similar sets fit into these categories although I have not been able to prove this fact and am looking for some help proving it or a counterexample in another category (most likely non rectifiably connected sets that are not Jordan arcs). I shall state the categories and give an example from each to give some clarification.

Non rectifiably connected Jordan arcs (Koch arc)

Rectifiably connected sets (Sierpenski Gasket)

Totally disconnected sets (C x C : a 2 dimensional cantor set)

Thank you for your help. I apologise if I have been unclear anywhere, I will try and clarify any points that need it.