  We investigate the effects of unparticles on Compton scattering, e gamma -> e
gamma based on a future e^+e^- linear collider such as the CLIC. For different
polarization configurations, we calculate the lower limits of the unparticle
energy scale Lambda_U for a discovery reach at the center of mass energies
sqrt(s)=0.5 TeV- 3 TeV. It is shown that, especially, for smaller values of the
mass dimension d, (1 <d <1.3), and for high energies and luminosities of the
collider these bounds are very significant. As a stringent limit, we find
Lambda_U>80 TeV for d<1.3 at sqrt(s)=3 TeV, and 1 ab^(-1) integrated luminosity
per year, which is comparable with the limits calculated from other low and
high energy physics implications.
