  We compute the spectral distortions of the Cosmic Microwave Background (CMB)
created by an exotic process that extracts or injects photons of a particular
frequency into the CMB. Such signatures are a natural prediction of a class of
composite dark matter models characterized by electrically neutral states but
with non-zero higher order electromagnetic moments. We consider a simplified
model where dark matter exists as a two state system separated by a fixed
transition frequency, which can range from radio waves to gamma rays. The
electromagnetic transitions between the two states due to CMB photons give rise
to thermal distortions, namely, the $\mu$-type distortion in the redshift range
$10^5\lesssim z \lesssim 2\times 10^6$ and the $y$-type distortion as well as
non-thermal distortions at redshifts $z \lesssim 10^5$. The nature of spectral
distortions depends sensitively on the dark matter transition frequency and the
strength of couplings of dark matter with visible sector particles as well as
its self-interactions, thus opening a new window to probe the nature of dark
matter. Non-thermal distortions have unique spectral shapes making them
distinguishable from the standard $\mu$ and $y$-type distortions and
potentially detectable in the next-generation experiments such as Primordial
Inflation Explorer (PIXIE). We also find that the spectral distortion limits
from the COsmic Background Explorer/Far-Infrared Absolute Spectrophotometer
(COBE/FIRAS) already give a constraint on the electromagnetic coupling of dark
matter which is three orders of magnitude stronger compared to the current
direct detection limits for $\sim$ MeV mass dark matter with transition energy
in $\sim 1$-$10$ eV range.
