  We study the fractional quantum Hall phases of a pseudospin-1/2 Bose gas in
an artificial gauge field. In addition to an external magnetic field, the gauge
field also mimics an intrinsic spin-orbit coupling of the Rashba type. While
the spin degeneracy of the Landau levels is lifted by the spin-orbit coupling,
the crossing of two Landau levels at certain coupling strengths gives rise to a
new degeneracy. We therefore take into account two Landau levels, and perform
exact diagonalization of the many-body Hamiltonian. We study and characterize
the quantum Hall phases which occur in the vicinity of the degeneracy point.
Notably, we describe the different states appearing at the Laughlin filling,
\nu=1/2. While for this filling incompressible phases disappear at the
degeneracy point, denser systems at \nu=3/2 and \nu=2 are found to be clearly
gapped. For filling factors \nu=2/3 and \nu=4/3, we discuss the connection of
the exact ground state to the non-Abelian spin singlet states, obtained as the
ground state of k+1 body contact interactions.
