  We address the important question of how to characterize the normal state of
fermionic superfluids under the influence of a strong effective magnetic field,
implemented through rapid rotation or novel artificial field techniques. We
consider the effects of crossing from BCS to BEC and the role of non-condensed
pairs, or pseudogap effects. Using a simple extension of Gor'kov theory we
demonstrate how these pairs organize above the transition $T_c$ into precursors
of a vortex configuration, which are associated with distortions of the ideal
Abrikosov lattice. This non-uniform normal state appears to enable "Bose
condensation" in a field which is otherwise problematic due to the effective
one-dimensionality of Landau level dispersion.
