Probably I did not make the question very clear. I explained it in the comment to the answer given by Dietrich.

I have read section 8 of Onishchik's book before I am posing the question. So it is clear to me that there is some classification. What I am asking is whether there is a classification using highest weight of the representation with respect to the restricted root system of the real Lie algebra.

@Artie: I agree. It seems rational relates to open subsets of affine space in algebraic geometry.

@Artie: Maybe you are right, but that's not what I want to know. The question is modified.

Krasner's Lemma can be used to prove that every local field of characteristic zero comes from completion of a number field. Therefore it answers the question.

It is known that a local field of characteristic zero is a finite extension of $Q_2$. This why I said that $K*/(K^*)^2$ has order 1,2,4 or 8 is incorrect. The question is whether the order argument is true for local fields coming from completions of number field? If every local field comes from the completion of a number field, then of course the order argument is incorrect.