If I remember correctly, time actually slows down as you approach c (speed of light in a vacuum). I believe that was one of the implications of special relativity. There's a reasonably simply equation for calculating it, which I toyed with during high school physics while everyone else was trying to figure out what the heck an electron was.
You remember correctly. As I recall, and as I was trying to explain earlier, the equation basically comes down to: Time will be distorted in such a way that light appears to be moving at the same speed regardless of the velocities of the vantage points in question. I've played around with it myself off-and-on for years since I first read Carl Sagan.