  We introduce a microscopic model on the honeycomb bilayer, which in the
small-momentum limit captures the usual (quadratic dispersion in kinetic term)
description of bilayer graphene. In the limit of strong interlayer hopping it
reduces to an effective honeycomb monolayer model with also third neighbor
hopping. We study interaction effects in this effective model focusing on
possible superconducting instabilities. We find d_{x^2-y^2} superconductivity
in the strong coupling limit of an effective tJ-model-like description that
gradually transforms into d + id time-reversal symmetry breaking
superconductivity at weak couplings. In this limit the small momentum order
parameter expansion is (k_x + i k_y)^2 [or (k_x - ik_y)^2] in both valleys of
the effective low-energy description. The relevance of our model and
investigation for the physics of bilayer graphene is also discussed.
