Among the reasons that $\mathbf{C}_p$ is hard to "visualize" are the it is totally disconnected (as is $\mathbf{Q}_p$) and it is not locally compact. The lack of local compactness means, for example, that you can't put a nice measure on $\mathbf{C}_p$. Many people these days instead work on the Berkovich affine line $\mathbf{A}_p^{Berk}$ or the associated Berkovich projective line $\mathbf{P}_p^{Berk}$. The Berkovich line is a topological space that
- contains a copy of $\mathbf{C}_p$ as a topological space
- is (simply) connected;
- is locally compact.
So people do measure theory, and even harmonic analysis, on Berkovich spaces. You can find a brief introduction, with some pictures, in my book The Arithmetic of Dynamical Systems, Springer, Section 5.10. For a more complete introduction, there's a great new book by Baker and Rumely, Potential Theory and Dynamics on the Berkovich Projective Line, American Mathematical Society, 2010.
Final comment: The fact that $\mathbf{C}_p$ is not spherically complete, which was mentioned by Neil Strickland, plays a role in Berkovich space. More precisely, it leads to some extra points that are needed to make Berkovich space complete.