Two people (so far) have tried to provide some sensible information, but I think the question itself is too loosely formulated to have a real answer. What you are looking at is the group of rational points of an algebraic group of rank 1 over a certain type of infinite field, which could be of characteristic 0 or not. This group itself is just an abstract group, which doesn't immediately have a definite "Weyl group" attached. It makes more sense to start with the algebraic group $G=\mathrm{SL}_2$, which is defined and split over any prime field. (The language of group schemes is appropriate here.) Over any field $F$, the group of rational points is just the usual group of $2 \times 2$ matrices of determinant 1 over $F$. It makes no difference for your question what $F$ is.

In $G$ there is a single well-defined (up to isomorphism) *Weyl group* $N_G(T)/T$ which just has order 2, as described concretely by Mrc Plm in the split case. All maximal tori of $G$ are conjugate as well as self-centralizing, so the choice doesn't matter.

However, if $F$ fails to be algebraically closed, the resulting abstract group
$\mathrm{SL}_2(F)$ is more challenging because $G$ typically has more than one class of maximal tori defined over $F$. In your rank 1 case it's not so bad because there are just two possibilities, the "split" and "anisotropic" (or "compact") types of maximal tori. The structure of the normalizer modulo centralizer of each type has to be looked at case-by-case in terms of the "absolute" Weyl group. In higher ranks the situation gets more complicated. Anyway you have to take account of the variation of structure in the points of $G$ over an arbitrary field.

Again I'd emphasize that the basic theory coming from study of algebraic groups doesn't depend on which field of definition $F$ (even finite) you are ultimately interested in, though of course the end results about rational points depend heavily on that field. There are somewhat different expositions of all this in standard textbooks, but the idea is always the same in your limited situation.

`$F$`

is given, though the tag p-adic-groups is suggestive. Usually "the" Weyl group is defined for an algebraic group over an arbitrary algebraically closed field, relative to any given maximal torus (all such tori being conjugate). It would help to know what kind of textbook source you are starting with, since the question is certainly "basic and elementary" once the context is clear. But more precision is needed if you start to look at fields of definition which are not algebraically closed. – Jim Humphreys May 6 '12 at 13:53