I'm wondering if there is some generic topology that can be put on any field of characteristic zero which is similar to those induced by a norm on the field. I know that for vector spaces you can take the terminal topology induced by the family of all semi-norms on the space, but I don't think that this topology is also a field topology in general. Can one take the join of all topologies induced by a norm on the field and hope for a field topology?
Any locally compact field, which is not discrete, is a non-archimedean or archimedean field.
Too long for a comment.
You might find useful the two books of S.Warner [Topological fields; topological rings], and the book N.Shell, Topological fields and near valuation. In Chap. 3 (The lattice of ring topologies) you find that the field (resp. ring, group) topologies form a complete lattice (with the same Sup and different meets).
On a algebraically closed field $K$ of characteristic 0, consider the collection of all involutions (equivalently, the set of all real closed subfields $R$ whose algebraic closure $R(i)$ in $K$ is $K$). Each such element gives, like in the case of the real numbers inside the complex numbers, a absolute value (with values in $R$) on $K$ whose topology is that of $R^2$. The sup of all such topologies is a "intrinsic" field topology on $K$ (quite analogue to the finest locally convex topology on a real vector space), but I know no explicit study of it (is this infinite sup of minimal field topologies discrete? It should be not reducible to a finite sup of such valuation topologies, hence not locally bounded).
In general, embed a field (of characteristic 0) in its algebraic closure (or a formally real field in its real closures, one for each ordering, and consider the sup of all order topologies).