Time? Our "consciousness moves along it" because we are shrouded (for training purposes?) in a three dimensional apparatus. Consider this from Christopher Langan. Cognizance processes information while time processes space. "...we can speak of time and space as equivalent to cognition and information with respect to the invariant semantic relation processes, as in "time processes space" and "cognition processes information"....reality (is) to be described as an autological (self-descriptive, self-recognizing/self-processing) system."
The biggest problem with modern day physics is that they're trying so desperately hard to prove naturalistic philosophy true instead of just working with the given evidence. That dark matter isn't going to show up and save all your theories no matter how hard you try. No matter how complex and clever a theory you create (hello, Steven Hawking) you cannot make something come from nothing.
And the idea that time is just another dimension is like saying cake is just another car part; its a different category. Humans enjoy both cake and cars, but they aren't somehow connected thereby. Similarly, people perceive dimensions and time.
Time is what is needed to cross space, from where you are to where you need to be, have to be, want to be. Time is how you measure how long you must wait to change what you would rather not do for what you would do with joy forever.
If you were where you wanted to be and didn't have to leave, doing what you loved and didn't have to stop, where would time go?
It would make you and everything around you older. Time is external, not perception. Our perceptions can make time seem to change (waiting in line vs eating wonderful food), but time is objective.