Shooting video underwater is very different than normal, above water, photography. The challenges are low visibility, particles in the water, loss of color and more..
While a video light is advised in every type of video shooting, it is an absouloute must in underwater videography. It is very easy to manipulate and enhance underwater still images but it is not so simple to enhance a video, this is why a video light is advised for every underwater shoot.
Here are the main advantages of using a video light underwater:
ADD LIGHT - The water itself and the floating particles in them absorb the light from the sun causing much lower light underwater that above even in sunny days. A video light simply adds synthetic light to the image.
ADD COLORS - The water acts as a blue filter , absorbing reds , yellow and warm colors causing images to be blueish and greyish. By using a video light, you would add a broad spectrum light , simliar to the sun close to the object causing the real colors to be reflected back to your camera. The result is vibrant color photos.
ASSIST FOCUS -The water , lighting and particles might make it more difficult for the cameras auto focus to work smoothly and accuratly , a video light can help significantly.
HALOGEN / LED
In the past, halogen bulbs were the most popularly used lights but today when the improvement of LED lights, most of the new lights are using LED. LED is more efficient and therefore has more battery life, generates less heat and is far more reliable. The downsides are that it is currently more expensive and it's light is not as "real" in terms of its spectrum as Halogen lights.
FLOOD / SPOT - BEAM ANGLE
A wide beam angle will spread the light over a large surface thus reducing the amount of light at a given point. A narrow beam angle will focus the light to a small circle with a greater intensity. As such, dive lights are usually narrow beamed to allow searching for objects and lighting to a distance to show the way. Video lights are usually with a wide beam to light as much of the frame as possible. Many modern lights today offer a dual set of LED's to allow switching between Flood and Spot underwater. Some lights like the Light and Motion GoBe allow switching light heads above water for different uses.
Generally , yes, it is possible but bear in mind that video lights , even very powerful ones with 2000 lumens are not nearly as powerful as a strobe. A rule of thumb is a basic GN 20 strobe to be equivalent to 6000-8000 Lumens of light.
As such , if your primary goal is shooting stills, a strobe is much more recommended. You are better off having a basic strobe and basic video light than a powerful video light only.
Stills shot with Light and Motion SOLA video 1200 (taken with Nikon P7000)