  We use Coulomb branch indices of Argyres-Douglas theories on $S^1 \times
L(k,1)$ to quantize moduli spaces ${\cal M}_H$ of wild/irregular Hitchin
systems. In particular, we obtain formulae for the "wild Hitchin characters" --
the graded dimensions of the Hilbert spaces from quantization -- for four
infinite families of ${\cal M}_H$, giving access to many interesting geometric
and topological data of these moduli spaces. We observe that the wild Hitchin
characters can always be written as a sum over fixed points in ${\cal M}_H$
under the $U(1)$ Hitchin action, and a limit of them can be identified with
matrix elements of the modular transform $ST^kS$ in certain two-dimensional
chiral algebras. Although naturally fitting into the geometric Langlands
program, the appearance of chiral algebras, which was known previously to be
associated with Schur operators but not Coulomb branch operators, is somewhat
surprising.
