Thanks for all the replies so far, there's been some good points made.
I feel like the current -max HP & 3 death saves makes PCs dieing too rare. In our couple of campaigns so far, we have had no deaths (but we have had injuries, which we roll for at zero hp).
I quite like the -ve half max HP idea, although I feel like that would still be too generous at higher levels (eg: fighter with 200HP, so -100 to die -... is that ever really going to be a serious risk?)
I suppose once you start tracking -ve HP, that all by itself might be enough to avoid whack-a-mole style play- it eliminates the incentive to drop to zero then heal. You might as well heal while above zero, because there is no free damage soak at zero hp, and your PC might lose a round or two while at zero.
Hmmm. Maybe that is the simplest solution? Do others already employ this approach?