  The localization of fast radio bursts (FRBs) has been hindered by the poor
angular resolution of the detection observations and inconclusive
identification of transient or variable counterparts. Recently a $\gamma$-ray
pulse of $380$ s duration has been associated with the fast radio burst FRB
131104. We report on radio-continuum imaging observations of the original
localization region of the FRB, beginning three days after the event and
comprising 25 epochs over 2.5 yr. Besides probabilistic arguments that suggest
that the association between the $\gamma$-ray transient and the FRB is not
compelling, we provide upper limits on a putative radio afterglow of this
transient that are at odds with standard models for its progenitor. We further
report the discovery of an unusual variable radio source spatially and
temporally coincident with FRB 131104, but not spatially coincident with the
$\gamma$-ray event. The radio variable flares by a factor of $3$ above its long
term average within $10$ d of the FRB at 7.5 GHz, with a factor-of-$2$ increase
at $5.5$ GHz. Within our observations, the variable has persisted with only
modest modulation and never approached the flux density observed in the days
after the FRB. We identify an optical counterpart to the variable. Optical and
infrared photometry, and deep optical spectroscopy, suggest that the object is
a narrow-line radio AGN.
