How far does lighting travel in water?

Answer 1

Not very far. The charge the lightning carries is disbursed in a short distance in water, and a shorter distance still in salt water. A few meters is about all that is required for the current density to drop to some small value. A few tens of meters maximum.

Answer 2

Through pure water it would travel indefinitely.

However, no water is pure in large quantities. Small impurities absorb a fraction of the light. (or reflect it) This eventually diminishes the light to a point where it is no longer visible. The length it can travel is dependent on what is in the water and the initial intensity of the light.

An in depth note:

If the light was being produced in a single wavelength and that wavelength matched the absorption spectrum of water the light would travel a very short distance. The length would probably not be measurable.