We study the collective modes of the quantum Hall states in undoped bilayer graphene in a strong perpendicular magnetic and electric field. Both for the well-known ferromagnetic state that is relevant for small electric field E-perpendicular to and the analogous layer polarized one suitable for large E-perpendicular to, the low-energy physics is dominated by magnetoexcitons with zero angular momentum that are even combinations of excitons that conserve Landau orbitals. We identify a long-wavelength instability in both states, and argue that there is an intermediate range of the electric field E-c((1)) < E-perpendicular to < E-c((s)) where a gapless phase interpolates between the incompressible quantum Hall states. The experimental relevance of this crossover via a gapless state is discussed.