  We discuss the possible relevance of complex codimension-two extremal
surfaces to the the Ryu-Takayanagi holographic entanglement proposal and its
covariant Hubeny-Rangamani-Takayanagi (HRT) generalization. Such surfaces live
in a complexified bulk spacetime defined by analytic continuation. We identify
surfaces of this type for BTZ, Schwarzschild-AdS, and Schwarzschild-Lifshitz
planar black holes. Since the dual CFT interpretation for the imaginary part of
their areas is unclear, we focus on a straw man proposal relating CFT entropy
to the real part of the area alone. For Schwarzschild-AdS and
Schwarzschild-Lifshitz, we identify families where the real part of the area
agrees with qualitative physical expectations for the appropriate CFT entropy
and, in addition, where it is smaller than the area of corresponding real
extremal surfaces. It is thus plausible that the CFT entropy is controlled by
these complex extremal surfaces.
