How much of the theory of C*-algebras holds when the complex numbers are replaced by different (algebraically closed) field (possibly with a distinguished ordered subfield that satisfies the same properties as the reals)?

Examples of some fields for which I would be interested in knowing if there are known analogs include (but aren't limited to):

- Surreals.
- Algebraic Closure of Surreals.
- The uncountable algebraic closed field of characteristic $p$.
- The p-adics, $\mathbb{Q}_p$.
- The algebraic closure of the p-adics, $\mathbb{C}_p$.

I realize this is a slightly ill-defined question, however it seems probable that it has none the less been worked out and so I would appreciate any relevant references.