  In this article we introduce conformal Bach flow and establish its
well-posedness on closed manifolds. We also obtain its backward uniqueness. To
give an attempt to study the long-time behavior of conformal Bach flow,
assuming that the curvature and the pressure function are bounded, global and
local Shi's type $L^2$-estimate of derivatives of curvatures are derived.
Furthermore using the $L^2$-estimate and based on an idea from \cite{St13} we
show Shi's pointwise-estimate of derivatives of curvatures without assuming
Sobolev constant bound.
