The speed of light is fixed. The reason a moving clock goes slower is the distance increases. Or more accurately space time curves.
Speed = Time/Distance
If speed can stay fixed so long as the Time and Distance increase at the same proportion.
It is an oddity that in all reality - what we see a straight line is not straight. And that curved lines can be straight. For instance - the moon is circling the earth in a straight line through space time. I know - I “kinda” understand it. Sort of. But yeah - the universe does not like to keep it simple.
As for why we eperiance
100% possible - but to you it would feel like the time passed normally. There 2 ways to do this. Travel as close to the speed of light as possible OR - expose yourself to massive gravity fields. As an FYI the book “tomorrow war” deals with the former and the film “interstellar” the second.
The idea that time moves slowly for those in VERY high gravity is well known. But here is a brain killer.
If I were to jump into a back hole - you would not see me die. My time would slow to the point where I would appear to freeze to you. And you would die long before I would. As my saw me move at billionth of a millimetre a year. Me on the other hand - I’d see you die. And then as I sank deeper it is 100% possible I’d see the galaxy die. And the even the universe. As the amount of time dilation is not known.