Question

If you were to look at a clock while moving away at the speed of light, what would you see the clock do?

Answer 1

The clock you are looking at WILL NEVER TICK!!

Explanation:

This a direct consequence of what is called time dilation.

Normally the explanation is given as follows: Intuitively, the light that left the clock will never reach you! But this explanation is incorrect, because special relativity clearly says that no matter what your speed the speed of light will be still the same, so regardless of you travelling at the speed of light, the speed of light coming out of the clock still would catch up to you!

The correct explanation would be as follows:

According to Special Relativity the time interval `\Delta T'` as observed by an observer is related to the stationary observer's corresponding time interval `\Delta T` as follows
`Delta T'= \gamma \Delta T`, where `\gamma= 1/{\sqrt{v^2-c^2}}` if the clock moves at the speed of light(`c`), `\gamma=\infty`, therefore any time interval in the stationary observers watch will correspond infinite interval in the moving clock's frame, ie the hands of the clock never move! The clock looks still!

PS: You asked about an observer moving at the speed of light, I have described it for a clock moving at `c` and a stationary observer, a little thought will tell you that both the scenarios are completely equivalent!

I know this could get a little confusing, don't hesitate ask anything that you don't understand!