  We demonstrate that the theoretical description of current experiments of
quasi-2D Fermi gases requires going beyond usual 2D theories. We provide such a
theory for the highly spin-imbalanced quasi-2D Fermi gas. For typical
experimental conditions, we find that the location of the recently predicted
polaron-molecule transition is shifted to lower values of the vacuum binding
energy due to the interplay between transverse confinement and many-body
physics. The energy of the attractive polaron is calculated in the 2D-3D
crossover and displays a series of cusps before converging towards the 3D
limit. The repulsive polaron is shown to be accurately described by a 2D theory
with a single interaction parameter.
