  While the standard construction of the S-matrix fails on Anti-de Sitter (AdS)
spacetime, a generalized S-matrix makes sense, based on the hypercylinder
geometry induced by the boundary of AdS. In contrast to quantum field theory in
Minkowski spacetime, there is not yet a standard way to resolve the
quantization ambiguities arising in its construction. These ambiguities are
conveniently encoded in the choice of a complex structure. We explore in this
paper the space of complex structures for real scalar Klein-Gordon theory based
on a number of criteria. These are: invariance under AdS isometries, induction
of a positive definite inner product, compatibility with the standard S-matrix
picture and recovery of standard structures in Minkowski spacetime under a
limit of vanishing curvature. While there is no complex structure that
satisfies all demands, we emphasize two interesting candidates that satisfy
most: In one case we have to give up part of the isometry invariance, in the
other case the induced inner product is indefinite.
