  To gain insight into how bulk locality emerges from the holographic conformal
field theory, we reformulate the bulk to boundary map in as local a way as
possible. In previous work, we carried out this program for Lorentzian AdS, and
showed the support on the boundary could always be reduced to a compact region
spacelike separated from the bulk point. In the present work the idea is
extended to a complexified boundary, where spatial coordinates are continued to
imaginary values. This continuation enables us to represent a local bulk
operator as a CFT operator with support on a finite disc on the complexified
boundary. We treat general AdS in Poincare coordinates and AdS3 in Rindler
coordinates. We represent bulk operators inside the horizon of a BTZ black hole
and we verify that the correct bulk two point functions are reproduced,
including the divergence when one point hits the BTZ singularity. We comment on
the holographic description of black holes formed by collapse and discuss
locality and holographic entropy counting at finite N.
