Interpreting "progress" in a different (perhaps more controversial!) way than in Tony Scholl's answer, one could also mention that Langlands (followed by Kottwitz and perhaps others) has introduced a hypothetical group $L_k$ which should be even bigger than $W_k$, normally referred to as the Langlands group, which should bear the same relationship to arbitrary automorphic forms as $C_k$ does to Grossencharacters.

It might help to remark that the algebraic Grossencharacters correspond geometrically to abelian varieties with CM, and so the algebraic envelope of $C_k$ (the associated pro-algebraic torus through which all algebraic Grossencharacters factor) is the Tannakian group of the category of motives over $k$ generated by the Tate motive together with the motives of all abelian varieties over $k$ which have CM defined over $k$.

The algebraic envelope of $W_k$ (which is now a non-commutative reductive pro-algebraic group) is the Tannakian group of the category of motives over $k$ generated by motives which are *potentially* CM, i.e. which become CM motives (i.e. belong to the category considered in the
preceding paragraph, i.e. are classified by an algebraic Grossencharacter on $C_l$) over some extension $l$ of $k$. This category contains all Artin motives, for example.

The algebraic envelope of $L_k$ should be the Tannakian group of the category of all motives over $k$.

So the problem of constructing $L_k$ can be thought of, from this point of view, as the problem of enlarging the category of motives so that one can make sense of motives with "non-integral Hodge grading" (i.e. has $h^{p,q}$ for $p$ and $q$ complex numbers
that are not necessarily integral); $L_k$ would then be (some version of) the Tannakian group of this category.

Going back to $C_k$, from this optic one would like to generalize the notion of CM abelian variety to include objects with non-integral Hodge gradings, which would give rise to non-algebraic Grossencharacters in the way that usual CM abelian varieties correspond to algebraic Grossencharacters.

And of course, for $W_k$, once wants to generalize potentially CM abelian varieties in the same way.