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It is well known that $\mathbb{C}$ and $\mathbb{C}_p$ are "algebraically" isomorphic (that is, ignoring the topology), but an isomorphism depends on the axiom of choice and there is no canonical way to relate and archimedean and a p-adic valuation. Still, doing so for all $p$ you have a field where you have an archimedean valuation, and can be equipped with a $p$-adic valuation for each $p$ which makes it complete, extending the archimedean and $p$-adic valuations of $\mathbb{Q}$ (or a finite extension thereof).

I was wondering if this can be done in order to build a field where a "product formula" for valuation holds $\prod_v |x|_v=1$, and which is complete for all valuations $|\cdot|_v$. Note that we cannot expect that $|x|_v=1$ for all $v$ except a finite set, or our field would be forced to be either an algebraic number field, either an algebraic function fields, by [Artin-Whaples, Axiomatic characterization of fields by the product formula for valuations]. But still, such a product could converge for all $x$.

I don't know if this may look like a "random question", but I was interested in understanding what are the limits in the study of statements about complex numbers via the reduction to $p$-adic equivalents, which sometimes happens in algebraic dynamics (and often can be made to work by just reducing to the dense set of algebraic numbers and using density).

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