It depends a lot on what you mean when you say "absolute motion". Note that the unruh effect that some posters brought up does not detect "absolute motion" via the usual meaning.
A more-or-less standard definition of absolute motion is that there would be some experiment you could perform in a closed room without looking "out the windows" to determine the room's velocity.
Your question is really a little to vague to answer in any detail, unless you like very general answers like "detection of absolute motion would violate the principle of Lorentz invariance which is the foundation of relativity". If you want a more specific answer, you need to ask a more specific question, I'm afraid, one could imagine a lot of hypothetical ways in which Lorentz symmetry could be violated. Also, since experimental evidence strongly supports Lorentz invariance, the most likely candidates would be physics which almost preserves Lorentz invariance. It's unclear though what sort of scenario you are thinking about from your question.