What makes sense to me is to go a route similar to what you did. So, let's think of, just for now, a subset of the real numbers which survives two applications of . Something like . What about a set which survives three? What if we took now the subset of the complex numbers which has the real line, appended to the vertical line and then for each we considered for all . Then, I feel that and . What if you added to each point of the a sequence converging to it, and to each one of those points append a, etc.
I don't feel this is correct though. Let me think about it.