I think that Einstein said that as matter accelerates it increases in mass, it shortens in length, and time slows down. That would seem to correspond to matter containing Higgs bosons (which give it mass) passing through the Higgs field, which slows it down, like moving through molasses. As speed increases it becomes more and more difficult for the matter to move through the Higgs field. The field offers more and more resistance, thus having the effect of increasing mass and requiring more and more energy to accelerate. Meanwhile the pressure of the Higgs field might tend to compress the matter in the direction of motion, like pushing a balloon through water; it would become more and more deformed, becoming flatter and flatter the harder and faster you push it through the water. Finally the increased resistance of the Higgs field might slow everything down; because of the increasing resistance of the Higgs field everything would move in slow motion, compared to something not moving so quickly through the field; thus making time slow down.
The problem with this theory is that in the theory of relativity, everything is relative, while the Higgs field would appear to be static. Thus if you launched a rocket from earth, things would appear to be compressed, slow down, etc., on the rocket as viewed from earth. However, as views from the rocket, the earth would appear to be compressed, slow down, etc. These are effects of relative motion, not absolute motion; so the Higgs field could not be some static field measuring absolute motion. The Higgs field would have to be relative to the observer, just as the relativity effects are.
