For simplicity, call the Weil group of a local nonarchimedean field $F\_v$ to be the following extension:
$$1\longrightarrow F^\times\_v\longrightarrow W\_{F\_v}\longrightarrow\text{Gal}(F\_v/\mathbb Q\_v)\longrightarrow 1.$$


1. Finding this definition to be lacking in terms of describing $\ell$-adic representations, Deligne modified it to what is known as the Weil-Deligne group, $WD\_{F\_v}=W\_{F\_v}\ltimes\mathbb G\_a.$
2. Langlands in the Corvallis proceedings suggests that one should take instead $W'\_{F\_v}=W\_{F\_v}\times SL(2,\mathbb C)$.
3. Kottwitz (and Arthur, citing him) later writes that it is known that (the Langlands group) should be $L\_{F\_v}=W\_{F\_v}\times SU(2,\mathbb R)$, to give a natural bijection between the irreducible n-dimensional complex representations of $L\_{F\_v}$ irreducible admissible representations of $GL\_n(F\_v)$.


It seems that (3) is the accepted form of the Langlands group, is (2) then no longer relevant? Is (1) now subsumed under (3), so that one should really be thinking of the latter?


As a follow up question, why is it that $L\_F$ *should be* an extension of $W\_F$ by a compact group (Kottwitz)? Or even a reductive, proalgebraic group (Langlands)?



Which Weil group over a $p$-adic field?
First, (2) and (3) are functionally equivalent. What matters are not the group themselves, but their categories of finite-dimensional complex representations (continuous in case (3), algebraic in the case (2)). But the two are the same: that's the simplest case of the famous Weyl's unitary trick. So working with (2) or (3) is just a question of taste, or of context. (2) is more natural in the context of Tannakian formalism, where we work with pro-algebraic group, while (3) is simpler.


Second, yes (1) is subsumed in (3). To be precise the category of Weil-Deligne representations of (1) is equivalent in a simple way to the category of representations of (3), so we can forget about (1). Most people work with (3) nowadays. 


