"The weird part: the resulting field is isomorphic to the complex numbers equipped with a weird metric. Using the axiom of choice. 😬"
Is this something someone has actually proven or is it just something that people assume?
Because when I took the seminar on this (circa 1982) , the word then was that we know very little about Ωₚ (the ultimate algebraically closed Cauchy-completion of 𝐐ₚ), that so far as we know, it has nothing to do with the complex numbers and Koblitz's book (what we were using) had nothing further to say,
and it seems to me there's rather a lot of work to do to axiom-of-choice a consistent metric out of this. But maybe somebody's done it in the last 30 years.