The field of $p$-adic numbers is regarded as more or less independent on the reals, with "strange" elements like square roots of negative numbers etc. Yet, there is this section on Wikipedia that to me seems like giving away some of their exoticness.
As I understand it, $\mathbb Q_p$ can be algebraically extended to form $\overline{\mathbb Q}_p$ which is not Cauchy complete, but this can be completed to form $\mathbb C_p$. This is both complete and algebraically closed, and as both transformations are field extensions, it essentially still contains the elements of $\mathbb Q_p$ in some form.
Next, it is stated that $\mathbb C_p$ and $\mathbb C$ are isomorphic as rings. How is this different from being isomorphic as fields? I suppose they are also isomorphic as commutative rings (as both are fields), but why would division be different? As it is isomorphic for multiplication, is there actually still some leeway for division of (respective) irrationals to differ?
Next it says that we can therefore regard $\mathbb C_p$ as $\mathbb C$ endowed with an exotic metric. However, I don't really consider metric a part of a field (or a ring) in the first place, unless we are talking about metric spaces specifically. After all, I can equip $\mathbb C$ with a lot of "exotic" metrices, so what is the difference and how is it actually related to the previous statement?
Lastly, this implies to me that any $p$-adic number can be sensibly regarded as a complex number (assuming AC). If that is indeed the case, is it safe to say that for example $\sqrt{-1}$ (when existing) in any $\mathbb Q_p$ "is" actually one of $i$ or $-i$, i.e. the image of such an isomorphism? If not, is there a known example of such a "truly exotic" $p$-adic number?
