I hope this is appropriate for mathoverflow. Understanding $\mathbb{C}_p$ has always been something of a stumbling block for me. A standard thing to do in number theory is to take the completion $\mathbb{Q}_p$ of the rationals with respect to a $p$-adic absolute value. The resulting field is then complete, but has no good reason to be algebraically closed. You can take its algebraic closure, but that is not complete, so then you take the completion of that, and get a field which is both complete, and algebraically closed, denoted by $\mathbb{C}_p$.

I understand that it is a reasonable desire to have a field extension of $\mathbb{Q}_p$ that is both complete and algebraically closed; my trouble, however, is getting some sort of grasp on how to picture this object, and to develop any intuition about how it is used. Here are my questions; I'd imagine the answers are related:

1. Am I even supposed to be able to picture it?
2. Is there some way I ought to think of a typical element?
3. Is it worth it, in terms of these goals, to look at the proofs of the assertions in my first paragraph?
4. How is $\mathbb{C}_p$ typically used? (this question may be too vague, feel free to ignore it!)

Please feel free to answer any or all of these questions.