  Using the chiral algebra bootstrap, we revisit the simplest Argyres-Douglas
(AD) generalization of Argyres-Seiberg S-duality. We argue that the exotic AD
superconformal field theory (SCFT), $T_{3,{3\over2}}$, emerging in this duality
splits into a free piece and an interacting piece, T_X, even though this
factorization seems invisible in the Seiberg-Witten (SW) curve derived from the
corresponding M5-brane construction. Without a Lagrangian, an associated
topological field theory, a BPS spectrum, or even an SW curve, we nonetheless
obtain exact information about T_X by bootstrapping its chiral algebra,
chi(T_X), and finding the corresponding vacuum character in terms of Affine
Kac-Moody characters. By a standard 4D/2D correspondence, this result gives us
the Schur index for T_X and, by studying this quantity in the limit of small
S^1, we make contact with a proposed S^1 reduction. Along the way, we discuss
various properties of T_X: as an N=1 theory, it has flavor symmetry
SU(3)XSU(2)XU(1), the central charge of chi(T_X) matches the central charge of
the bc ghosts in bosonic string theory, and its global SU(2) symmetry has a
Witten anomaly. This anomaly does not prevent us from building conformal
manifolds out of arbitrary numbers of T_X theories (giving us a surprisingly
close AD relative of Gaiotto's T_N theories), but it does lead to some open
questions in the context of the chiral algebra / 4D N=2 SCFT correspondence.
