How do migratory birds navigate?

Do they have some magnetic "biochemical" compass?
Maybe. Nice reviews in PNAS and Physics Today discuss the status of this question.

Here a few of the demanding physical requirements for this hypothesis of "biochemical magnetoreception":

• There is biochemical reaction similar to that shown below.

• The relative yield of C depends of the direction of magnetic fields of the order of 50 microtesla.
• Note that the corresponding electron Zeeman energies are more than 6 orders of magnitude smaller than the thermal energy, k_BT!
• The relevant competing reactions must occur on the same timescale as the Larmor precession of the electron spins associated with the 50 microtesla field. (1 microsecond)
• The relevant biomolecules must be aligned predominantly in one direction so the relevant hyperfine tensor has a well defined direction relative to the terrestial magnetic field.

What is the source of random magnetic fields that could destroy the coherence of the spin quantum states?
Spin isotropic interactions such as those associated with the dielectric fluctuations or exchange interactions with the environment cannot produce such interactions.
What about decoherence due to random nuclear spins?
Experimental studies have quantified this for qubits based on quantum dots.