Code available at https://github.com/thedevbc/stopwatch
Feel free to give some feedback.
The challenge with doing this in the browser is that the document can't be updated every millisecond. Hundreds of a second was possible, but it really didn't matter to the human eye. I decided to only show tenths of a second in the clock display, but then when recording values (laps) it is using time stamps which are accurate down to the millisecond.