  We employ an ab-initio structure search algorithm to explore the
configurational space of Bi in quasi two dimensions. A confinement potential
restricts the movement of atoms within a pre-defined thickness during structure
search calculations within the minima hopping method to find the stable and
metastable forms of bilayer Bi. In addition to recovering the two known
low-energy structures (puckered monoclinic and buckled hexagonal), our
calculations predict three new structures of bilayer Bi. We call these
structures the $\alpha$, $\beta$, and $\gamma$ phases of bilayer Bi, which are,
respectively, 63, 72, and 83 meV/atom higher in energy than that of the
monoclinic ground state, and thus potentially synthesizable using appropriate
substrates. We also compare the structural, electronic, and vibrational
properties of the different phases. The puckered monoclinic, buckled hexagonal,
and $\beta$ phases exhibit a semiconducting energy gap, whereas $\alpha$ and
$\gamma$ phases are metallic. We notice an unusual Mexican-hat type band
dispersion leading to a van Hove singularity in the buckled hexagonal bilayer
Bi. Notably, we find symmetry-protected topological Dirac points in the
electronic spectrum of the $\gamma$ phase. The new structures suggest that
bilayer Bi provides a novel playground to study distortion-mediated
metal-insulator phase transitions.
