  We study the influence of angular momentum on quantum complexity for CFT
states holographically dual to rotating black holes. Using the holographic
complexity=action (CA) and complexity=volume (CV) proposals, we study the full
time dependence of complexity and the complexity of formation for two
dimensional states dual to rotating BTZ. The obtained results and their
dependence on angular momentum turn out to be analogous to those of charged
states dual to Reissner-Nordstr\"om AdS black holes. For CA, our computation
carefully accounts for the counterterm in the gravity action, which was not
included in previous analysis in the literature. This affects the complexity
early time dependence and its effect becomes negligible close to extremality.
In the grand canonical ensemble, the CA and CV complexity of formation are
linear in the temperature, and diverge with the same structure in the speed of
light angular velocity limit. For CA the inclusion of the counterterm is
crucial for both effects. We also address the problem of studying holographic
complexity for higher dimensional rotating black holes, focusing on the four
dimensional Kerr-AdS case. Carefully taking into account all ingredients, we
show that the late time limit of the CA growth rate saturates the expected
bound, and find the CV complexity of formation of large black holes diverges in
the critical angular velocity limit. Our holographic analysis is complemented
by the study of circuit complexity in a two dimensional free scalar model for a
thermofield double (TFD) state with angular momentum. We show how this can be
given a description in terms of non-rotating TFD states introducing
mode-by-mode effective temperatures and times. We comment on the similarities
and differences of the holographic and QFT complexity results.
