When looking through the topic here on MSE, one can find in particular the questions P adic numbers number theory and 4-adic numbers and zero divisors which boil down to the facts, that either the inverse limits for the construction of $\mathbb{Z}_2$ and $\mathbb{Z}_4$ agree or one cannot define a suitable norm for composite numbers in parallel with the $p$-adic norms, which would give something new by Ostrowski's Theorem https://proofwiki.org/wiki/Ostrowski%27s_Theorem .
Now the question, have metrics $d_k$ on $\mathbb{Q}$ been considered, which are for $k$ not prime not induced by a norm but which agree with the $p$-adic norm if $k$ is prime? Naively, then for $\mathbb{Q}$ equipped with the metric $d_k$, we could hope that the completion $\bar{\mathbb{Q}}^{d_k}$ gives something that looks like the $p$-adics even for composite numbers. I have no good idea for the definition of $d_k$ but would hope that maybe someone could point me to a reference, if this has already bin considered in the literature.
