  We explore the two holographic complexity proposals for the case of a 2d
boundary CFT with a conformal defect. We focus on a Randall-Sundrum type model
of a thin AdS$_2$ brane embedded in AdS$_3$. We find that, using the
"complexity=volume" proposal, the presence of the defect generates a
logarithmic divergence in the complexity of the full boundary state with a
coefficient which is related to the central charge and to the boundary entropy.
For the "complexity=action" proposal we find that the complexity is not
influenced by the presence of the defect. This is the first case in which the
results of the two holographic proposals differ so dramatically. We consider
also the complexity of the reduced density matrix for subregions enclosing the
defect. We explore two bosonic field theory models which include two defects on
opposite sides of a periodic domain. We point out that for a compact boson,
current free field theory definitions of the complexity would have to be
generalized to account for the effect of zero-modes.
