It seems several issues got tossed up and that is clouding the picture.
This is not about the merits of hydrogen - it is about a device and what it purports to do.
The energy needed for the hydrolysis of water uses more than the energy it creates (through the use of the hydrogen). As noted previously, if this were not the case you could build a perpetual motion machine (or we would all be driving cars that run on water*). Now, if the alternator was always putting out wasted amps and those were used, that may be a different subject. But an alternator consumes more power as it creates electricity (and the generator example was spot on to illustrate that). Moreover, your alternator is not 100% efficient so you are losing again there as well.
Now, there may be some merit to adding hydrogen to a petrol powered car - I haven't the foggiest if it would help or not. If it helps and you could get hydrogen to add, great. My only point, and I believe the point of several others, is that getting that hydrogen through hydrolysis powered by the vehicle itself will not add efficiency.
By the way, Honda will be selling a limited number of Hydrogen fueled cars in California I believe as soon as next year. The reason is emissions, not fuel efficiency. Of course, if the fuel is cheap and clean efficiency doesn't matter all that much.
*I know, I know, an inventor built such a car in the 50's and the oil companies, GM, Ford, Eisenhower had him killed and shelved the technology
