  In this second part of the study initiated in arxiv:1804.07410, we
investigate holographic complexity for eternal black hole backgrounds perturbed
by shock waves, with both the complexity$=$action (CA) and complexity$=$volume
(CV) proposals. In particular, we consider Vaidya geometries describing a thin
shell of null fluid with arbitrary energy falling in from one of the boundaries
of a two-sided AdS-Schwarzschild spacetime. We demonstrate how known properties
of complexity, such as the switchback effect for light shocks, as well as
analogous properties for heavy ones, are imprinted in the complexity of
formation and in the full time evolution of complexity. Following our
discussion in arxiv:1804.07410, we find that in order to obtain the expected
properties of the complexity, the inclusion of a particular counterterm on the
null boundaries of the Wheeler-DeWitt patch is required for the CA proposal.
