and "choosing" every element of the (uncountably large) bijection
I knew that part. (ANY pulling of uncountably large rabbits from hats ⟹ AoC)
What I was missing was how to deal with the question of how you map some z that's the limit of a sequence in ℂ, to something appropriate in ℂₚ, where, even if the sequence were entirely rational numbers, it might not even converge in ℂₚ because the metric there is totally different and so you won't even have anything to choose from.
And the answer is if your z is algebraically independent of all of the z's you've considered thus far, then you just pick ANYTHING AT ALL that's algebraically independent of all of the previous f(z)'s and the metrics can go fuck themselves.