the change in time is conjugated to the change in energy. This is one of the uncertainty conditions of quantum mechanics.
This formulation is experimentally verified by high energy physics measurements.
When quantum mechanics is combined with special relativisty as done in Dirac's equation, QED, QCD, and the Standard Model, it was detected that high energy particles approaching the speed of light live a lot longer than when they just travel at normal speed.
The conclusion is that when you travel at the speed of light, you cannot grow older. Your body clock does not move.