It was showed in "The Stone–Weierstrass Theorem for valuable fields" that the Stone–Weierstrass theorem holds for any topological field whose topology comes from an absolute value or a Krull valuation (these are exactly the fields with Type V topologies).
Since any topological field is a uniform space, given a compact Hausdorff space $X$ the algebra of continuous functions $C(X,F)$ can be endowed with the uniform convergence topology (which is also the compact–open topology).
With this in mind, we say that a topological field $F$ (except for $\mathbb{C}$) satisfies the Stone–Weierstrass theorem if for all compact Hausdorff spaces $X$ if $A\subset C(X,F)$ is a subalgebra which contains the constant functions and separates points, then $A$ is uniformly dense (in the sense above).
Are there any examples of topological fields satisfying this theorem outside the metric cases? Is there a characterization of when a field satisfies this theorem?
