  We study the spectrum of a bilayer of Kitaev magnets on the honeycomb
lattice, coupled by Heisenberg exchange in the quantum-dimer phase at strong
interlayer coupling. Using the perturbative Continuous Unitary transformation
(pCUT) we perform series expansion starting from the fully dimerized limit, to
evaluate the elementary excitations, reaching up to and focusing on the
two-triplon sector. In stark contrast to conventional bilayer quantum magnets,
and because of the broken $SU(2)$-invariance, encoded in the intralayer
directional compass-exchange, the bilayer Kitaev magnet is shown to exhibit a
rich structure of two-triplon scattering-state continua, as well as several
collective two-triplon (anti)bound states. Direct physical pictures for the
occurrence of the latter are provided and the (anti)bound states are studied
versus the stacking type, the spin components, and the exchange parameters. In
addition to the two-triplon spectra, we investigate a corresponding
experimental probe by evaluating the magnetic Raman-scattering intensity. We
find a very strong sensitivity of this intensity on the two-triplon
interactions and the scattering geometry, however a signal from the (anti)bound
states appears only in very close proximity to the continuum.
