4 Added another picture and more remarks on extensions.

ADDED: Though this answer is becoming rather long, I wanted to add another picture to expand on the points I made about unramified and totally ramified extensions above.

Here is a picture of $\mathbb{Z}_3$, which I made with the free software Blender; imagine it continuing indefinitely upward:

A top view of this object should be the previous picture; the actual elements of $\mathbb{Z}_3$ should be viewed as sitting "infinitely high up" on the branches of this tree. As you can see, this object splits into levels, indexed by $\mathbb{N}$, and on the $n$-th level there are $p^n$ "platforms" corresponding to the residues mod $p^n$. For $\mathbb{Q}_p$, the levels should be indexed by $\mathbb{Z}$.

Now what happens when one looks at an unramfied extension of degree $k$? The levels, which correspond to powers of the maximal ideal, should not change, so the levels are still indexed by $\mathbb{Z}$; but the amount of branching on each "platform" is now indexed by $\mathcal{O}_K/m=\mathbb{F}_{p^k}$. So instead of having $p$ branches coming out of each level, one has $p^k$.

On the other hand, what if we have a totally ramified extension of degree $k$? Now $\mathcal{O}_k/m=\mathbb{F}_p$, so there are still $p$ branches on each level. But because the uniformizer now has valuation $1/k$, we can view the levels as being indexed by $\mathbb{Z}[1/k]$ (if you like, the height of each platform is now $1/k$ rather than $1$).

So what is the upshot for $\mathbb{C}_p$? We can view it as a similar diagram, except the levels are indexed by $\mathbb{Q}$, and the branches coming off of an individual platform correspond to elements of $\overline{\mathbb{F}_p}$.

One nice thing about this picture is that one can actually build spaces like the one I've included in the picture---replacing the tubes in my picture with line segments---such that the elements of $\mathbb{Q}_p$ or some extension thereof are a subset of the space (living "infinitely far" from the part I've drawn), with the subspace topology being the usual topology on the local field. Furthermore, the construction is functorial, in that an embedding $K\hookrightarrow K'$ induces a continuous map of spaces. The distance between two points in the local field is then given by their "highest common ancestor" in this garden of forking paths.

(This picture is essentially a description the Berkovich spaces mentioned by Joe Silverman, though I am essentially a novice in that regard, so it's quite possible I've made some mistake; you should take this as a description of my intuition, not Berkovich's definition.)

3 Stressed an important point.

I'll suggest a way to get a hold on $\mathbb{C}_p$ in a "pictorial" way. It is supposed to be similar to viewing $\mathbb{C}$ as a plane acting on itself via rotations, scalings, and translations.

There's a usual picture of $\mathbb{Z}_p$, which looks like the thing below for $p=3$ (taken from the website of Professor Katrin Tent):

Here the outermost circle is all of $\mathbb{Z}_3$; the three large colored circles are the residue classes mod $3$, the smaller circles are the residue classes mod $9$, and so on. If you want to think about $\mathbb{Q}_p$, imagine this picture continued infinitely "upward," (e.g. this circle is accompanied by two others, inside some larger circle, accompanied by two others, etc.).

Now the operations of multiplication and addition do something very geometric. Namely, addition cyclically permutes the residue classes (of each size!) by some amount, depending on the coefficient of $p^n$ in the $p$-adic expansion of whatever $p$-adic integer you have in mind. Multiplication by a unit switches the residue classes around as you'd expect, and multiplication by a multiple of $p^n$ shrinks the whole circle down and sends it to some (possibly rotated) copy of itself inside the small circle corresponding to the ideal $(p^n)$.

Now zero has the $p$-adic expansion $0+0\cdot p+0\cdot p^2+\cdots$ and so it is the unique element in the intersection of the circles corresponding to the residue class $0$ mod $p^n$ for every $n$. So we have a way to think of zeroes of polynomials over $\mathbb{Q}_p$---namely, a Galois extension of $\mathbb{Q}_p$ is some high dimensional vector space $\mathbb{Q}_p^N$ (which you probably have a picture of from linear algebra) acted on by $\mathbb{Q}_p$, in a way that twists each factor of $\mathbb{Q}_p^N$ and permutes the factors of the direct sum, according to the Galois action. Ramified extensions add intermediate levels of circles between those that already exist, whereas unramified extensions add new circles. That the extension is algebraic means that there's some way to twist it about (using the previously described actions) to put any element at the $0$ point.

Totally ramified extensions add intermediate levels of circles between those that already exist, whereas unramified extensions add new circles. I think this point of view is a particularly appealing visualization.

Now, the algebraic closure of $\mathbb{Q}_p$ is some maximal element of the poset of these algebraic extensions---which is hard to visualize as it is not really "unique," but for the sake of a picture one might think of choosing embeddings $K\to K'$ for each $K'/K$, and then taking the union. Finally, think of the completion in the usual way, e.g. by formally adding limits of Cauchy sequences.

Trying to draw pictures of some finite algebraic extensions of $\mathbb{Q}_p$ might help, and figuring out what the actions by addition and multiplication are is a fun exercise. I hope this "word picture" is as useful for you as it is for me.

2 Edited as per Pete Clark's comments.

I'll suggest a way to get a hold on $\mathbb{C}_p$ in a "pictorial" way. It is supposed to be similar to viewing $\mathbb{C}$ as a plane acting on itself via rotations, scalings, and translations.

There's a usual picture of $\mathbb{Z}_p$, which looks like the thing below for $p=3$ (taken from the website of Professor Katrin Tent):

Here the outermost circle is all of $\mathbb{Z}_3$; the three large colored circles are the residue classes mod $3$, the smaller circles are the residue classes mod $9$, and so on. If you want to think about $\mathbb{Q}_p$, imagine this picture continued infinitely "upward," (e.g. this circle is accompanied by two others, inside some larger circle, accompanied by two others, etc.).

Now the operations of multiplication and addition do something very geometric. Namely, addition cyclically permutes the residue classes (of each size!) by some amount, depending on the coefficient of $p^n$ in the $p$-adic expansion of whatever $p$-adic integer you have in mind. Multiplication by a unit switches the residue classes around as you'd expect, and multiplication by a multiple of $p^n$ shrinks the whole circle down and sends it to some (possibly rotated) copy of itself inside the small circle corresponding to the ideal $(p^n)$.

Now zero has the $p$-adic expansion $0+0\cdot p+0\cdot p^2+\cdots$ and so lies deep it is the unique element in the middle intersection of the circles corresponding to the residue class $0$ mod $p^n$ for every $n$. So we have a way to think of zeroes of polynomials over $\mathbb{Q}_p$---namely, a Galois extension of $\mathbb{Q}_p$ is some high dimensional vector space $\mathbb{Q}_p^N$ (which you probably have a picture of from linear algebra) acted on by $\mathbb{Q}_p$, in a way that twists each factor of $\mathbb{Q}_p^N$ and permutes the factors of the direct sum, according to the Galois action. Ramified extensions add intermediate levels of circles between those that already exist, whereas unramified extensions add new circles. That the extension is algebraic means that there's some way to twist it about (using the previously described actions) to put any element at the $0$ point.

Now, the algebraic closure of $\mathbb{Q}_p$ is some maximal element of the poset of these algebraic extensions---which is hard to visualize as it is not really "unique," but for the sake of a picture one might think of choosing embeddings $K\to K'$ for each $K'/K$, and then taking the union. Finally, think of the completion in the usual way, e.g. by formally adding limits of Cauchy sequences.

Trying to draw pictures of some finite algebraic extensions of $\mathbb{Q}_p$ might help, and figuring out what the actions by addition and multiplication are is a fun exercise. I hope this "word picture" is as useful for you as it is for me.

1