  We analyze different holographic complexity proposals for black holes that
include corrections from bulk quantum fields. The specific setup is the quantum
BTZ black hole, which encompasses in an exact manner the effects of conformal
fields with large central charge in the presence of the black hole, including
the backreaction corrections to the BTZ metric. Our results show that Volume
Complexity admits a consistent quantum expansion and correctly reproduces known
limits. On the other hand, the generalized Action Complexity picks up large
contributions from the singularity, which is modified due to quantum
backreaction, with the result that Action Complexity does not reproduce the
expected classical limit. Furthermore, we show that the doubly-holographic
setup allows computing the complexity coming purely from quantum fields - a
notion that has proven evasive in usual holographic setups. We find that in
holographic induced-gravity scenarios the complexity of quantum fields in a
black hole background vanishes to leading order in the gravitational strength
of CFT effects.
