TITLE: How to picture $\mathbb{C}_p$?
QUESTION [63 upvotes]: 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:

Am I even supposed to be able to picture it? 
Is there some way I ought to think of a typical element?
Is it worth it, in terms of these goals, to look at the proofs of the assertions in my first paragraph?
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.

REPLY [11 votes]: Since there are already several very good answers, I just discuss question 4 (how is ${\mathbb C}_p$ typically used?) with one example of use which made a great impression on me when I learnt it, and made me think that ${\mathbb C}_p$ was something deep and serious, and not only a very amusing curiosity. This example is a theorem of Tate and Sen, which states that if $V$ is a finite-dimensional over $\mathbb{ Q_p}$ vector space with a continuous linear action of $G=$Gal$(\overline{\mathbb Q_p}/{\mathbb Q_p})$ (that is, $V$ is a $p$-adic representation of $G$), then the following are equivalent:
(1) $\dim_{\mathbb Q_p} (V \otimes {\mathbb C_p})^{G} = \dim_{\mathbb Q_p} V.$ (Here, G acts on $\mathbb{C_p}$ by extending by continuity its action on $\overline{\mathbb Q_p}$
and it acts on $V \otimes {\mathbb C_p}$ by acting on both factors.)
(2) The inertia subgroup of $G$ acts on $V$ through a finite quotient (in more knowledeable  words, $V$ is potentially unramified).   
To appreciate this theorem, it may be useful to solve for oneself the following elementary exercise: if in (1), ${\mathbb C}_p$ is replaced by $\overline {\mathbb Q_p}$, then (2) should be replaced by "$G$ acts on $V$ through a finite quotient". Somehow, replacing 
$\overline {\mathbb Q_p}$ by its completion allows (1) to see inside the group $G$
and detect the behaviour of the inertia subgroup in it. 
I believe that someone who understands the proof of this theorem has necessarily
a good understanding of $\mathbb{C}_p$, and this will be my answer to question 3 as well:
knowing the proof of the basic assertions on $\mathbb{C}_p$ given in the questions is a first step into a good understanding of that field and its elements, but won't take you very far. Learning the proof of the above theorem will let you get a much deeper look inside  $\mathbb{C}_p$ -- and in addition you will learn a nice result, which is a first step in the fundamental $p$-adic Hodge Theory.