One point that I don't think anyone has mentioned yet is that $\mathbb{C}_p$ is isomorphic (as an untopologised field) to $\mathbb{C}$. More generally, any two algebraically closed fields of the same characteristic and cardinality are isomorphic, if I remember correctly. Of course the proof is horrendously non-constructive, but the very definition of $\mathbb{C}_p$ is already horrendously non-constructive. So instead of worrying about what $\mathbb{C}_p$ is, you can instead worry about why $\mathbb{C}$ admits a $p$-adic metric with respect to which it is complete. I don't have anything to offer about that.

One point that I don't think anyone has mentioned yet is that $\mathbb{C}_p$ is isomorphic (as an untopologised field) to $\mathbb{C}$. More generally, any two uncountable algebraically closed fields of the same characteristic and cardinality are isomorphic, if I remember correctly. Of course the proof is horrendously non-constructive, but the very definition of $\mathbb{C}_p$ is already horrendously non-constructive. So instead of worrying about what $\mathbb{C}_p$ is, you can instead worry about why $\mathbb{C}$ admits a $p$-adic metric with respect to which it is complete. I don't have anything to offer about that.

[Corrected as per Johannes Hahn's comment]