  Warped conformal field theories in two dimensions are exotic nonlocal,
Lorentz violating field theories characterized by Virasoro-Kac-Moody symmetries
and have attracted a lot of attention as candidate boundary duals to warped
AdS$_3$ spacetimes, thereby expanding the scope of holography beyond
asymptotically AdS spacetimes. Here we investigate WCFT$_2$\,s using
\emph{circuit complexity} as a tool. First we compute the holographic volume
complexity (CV) which displays a linear UV divergence structure, more akin to
that of a local CFT$_2$ and has a very complicated dependence on the Virasoro
central charge $c$ and the $U(1)$ Kac-Moody level parameter $k$. Next we
consider circuit complexity based on Virasoro-Kac-Moody symmetry gates where
the complexity functional is the geometric (group) action on coadjoint orbits
of the Virasoro-Kac-Moody group. We consider a special solution to
extremization equations for which complexity scales linearly with ``time''. In
the semiclassical limit (large $c,k$, while $c/k$ remains finite and small)
both the holographic volume complexity and circuit complexity scales linearly
with $k$.
