  Fermionic superfluids can undergo phase transitions into different kinds of
normal regimes, loosely characterized by whether Cooper pairs remain locally
stable. If the normal phase retains strong pairing fluctuations, it behaves
like a liquid of vortices, which has been observed in cuprate superconductors.
We argue that analogous strongly correlated normal states exist in
two-dimensional neutral fermion liquids near unitarity, where superfluid is
destroyed by fast rotation. These states have non-universal properties, and if
they develop as distinct thermodynamic phases they can be characterized as
quantum Hall states of Cooper pairs. The formal analysis is based on a model
with SP(2N) symmetry that describes the quantum critical region in the vicinity
of a broad Feshbach resonance. We explore the pairing phase diagram and
demonstrate that the considered model has macroscopically degenerate bosonic
modes in the normal phase, to all orders in 1/N. It takes finite-range
interactions to lift this degeneracy, making the Abrikosov flux lattice of the
superfluid particularly susceptible to quantum melting.
