Pretty straitforward:
If a field has a metric in which it is complete can it have a metric in which it is not complete?
By metric I mean field norm of course



Is completeness of a field an algebraic property?
How about the algebraic closure of the $p$-adics $\mathbb{Q}\_p^{\mathrm{alg}}$.
This is not complete under the $p$-adic metric, but it is isomorphic as
a field to the complex numbers $\mathbb{C}$ which is complete under the
standard metric (as both fields are algebraically closed of characteristic
zero with the same transcendence degree over $\mathbb{Q}$
(assuming AC of course)).


But if you want the second field to have nonarchimedean norm, take
the $p$-adic complex field $\mathbb{C}\_p$
(the completion of $\mathbb{Q}\_p^{\mathrm{alg}}$).


