I imagine many folks here have read the Angry GMs treatise on this topic? http://theangrygm.com/fighting-spirit/
I think his solution is a bit overcomplicated for my needs. But I do want something other than the binary options currently presented. Jester's idea of increasing exhaustion level every time the character hits zero is very nice in my opinion. It gives meaning to having your bell rung/being stabbed. You start to suffer even if you're given some temporary relief by a spell or potion.
It's a much better explanation of hit points - they then become an indicator of the "fight" left in you, rather than a record of damage taken and actually maps much better to combats we see in the movies where opponents are worn down to a point where they start making mistakes and risk injury.
Thanks for pointing out the article robus.
Angry DM is dealing with a related, but different, and larger problem.
The article makes me want to change my rule to state that a character at 0 hp is still conscious but dying. They must make a death saving throw each round. Or gain a level of exhaustion. But they can still act. This gives the emergency state angry DM was talking about, but in a different way.
maybe even give PCs the option of falling unconscious to automatically stabilize?