  We consider the Hamiltonian for neutrino oscillations in matter in the case
of arbitrary potentials including off-diagonal complex terms. We derive the
'matter basis' Hamiltonian in terms of quantities one can derive from the
flavor basis Hamiltonian and its derivative, for an arbitrary number of
neutrino flavors. We make our expressions explicit for the two-neutrino flavor
case and apply our results to the situation where the Hamiltonian includes both
coupling to matter and to neutrinos, which describes neutrino propagation in
core-collapse supernovae. We show that the neutrino flavour evolution depends
on the mixing matrix derivatives involving not only the derivative of the
matter mixing angles but also of the phases. In particular, we point out the
important role of the phase derivatives, that appear due to the
neutrino-neutrino interaction, and show how it can cause an oscillating
degeneracy between the diagonal elements of the matter basis Hamiltonian. Our
results also reveal, that the end of the synchronization regime is due to a
rapid increase of the phase derivative, and identify the condition to be
fulfilled for the onset of bipolar oscillations involving both the off-diagonal
neutrino-neutrino interaction contributions and the vacuum terms.
