In addition to having far better heat transfer (8x vs. air) hydrogen reduces windage losses, eliminates oxidation of the insulation, and requires a higher voltage to produce a corona discharge than air.
Unfortunately, a fatal accident happened on Jan. 8, 2007 involving hydrogen cooling at an AEP (American Electric Power) plant. I think it is way too early to draw any conclusions from this newspaper article.
Advantages of hydrogen cooling:
1. Reduction of windage loss by 90%
2. Increase in machine output
3. Lower oxidation of insulation
Disadvantages
1. Since Hydrogen-air mixture of between 5/95 to 75/25 is explosive, the hydrogen content inside the casing must over 95%
2. Sealing system to prevent air ingress
3. Explosion proof stator frame (though this doesn't seem to work in many cases)
Because hydrogen is less dense than air it is not centrifuged out of the rotor as much as air. Helium also has good thermal properties and is inert, but it is twice as dense as hydrogen.
swn1, concerning "...centrifuged out of the rotor..."
There are a lot of air cooled units too.
The surface of the rotor ( the blocks that retain the copper bars) have inlet and outlet scoops that pump cooling gas into the passages in the copper bars to the center of the rotor and back out. Then there is a fan at either one end or both that circulates the heated gas from the gap space through coolers to remove the heat.
h2 is the good removal of heat due to losses, it is also to keep h2 purity to accepatable levels. h2 purity of less than 90% contribute or increase losses by 10%. So at higher pressures h2 performs removes heat efficiently.