You don't start bleeding until you run out of Hit Points (which represent dodging, morale, etc). Then you start taking damage to your Life Points, which represent slashes, broken bones, and burnt flesh. (You've seen this idea before.)
Hp restores itself to full with a good night's rest.
You can restore your "rerolled HD-worth" of hp by listening to an inspiring speech. (So, for example, all level 2 magic-users would regain 2d4 hp). This requires a Charisma check by whoever is doing the inspirin'. A failed check means everyone only regains the minimum amount (2hp for our level 2 magic-user), and a superb result means that everyone gains an few bonus hp. This takes 1 minute.
Alternatively, if a PC is dying and hasn't benefited from the inspiring speech that day, one of their friends can slap them and cry "Live, damn you! Don't go into the light!" This takes 2 rounds and the PC is healed for 1d6 HP. If you have two friends beating on your chest and begging you to open your eyes, this only takes 1 round. A PC can inspire 1/day and be inspired 1/day.
Lastly, a gulp of hard liquor will restore 1d6 hp. This takes 1 round and can be used 1/day. This can also be used to revive dying characters, if you pour it down their throat.
All of the above can heal hp as normal, but cannot bring Life Points any higher than 1 (though they heal negative lp normally).
Additionally, all magical healing has a 5% chance to give you a random mutation (choose your favorite table). Also magic healing makes you fat. Everyone knows that.
Magical healing giving mutations is probably my favorite part of this. It means that magical healing will no longer be a first resort. Is the risk worth the healing?
Also, the PCs can heal themselves to full with a meal. When should we break for lunch? And I like this because it makes PCs in a dungeon seem more like explorers with real biological needs instead of video game characters that run for hours without food or water. If I could give a mechanical benefit to PCs for taking bathroom breaks I would.
Inspiration rules also make Cha more desirable. You might need it to save someone's life.
The rules about reviving dying characters add a small wrinkle, too. Do you want to drink your whiskey now or save it for when you are dying?
This system also takes the burden of healing off the cleric (if you swing that way) and/or makes the party less dependent on healing potions. Which is nice, if you are trying to run a low-magic campaign.
This is my own take on someone else's rules, but I can't remember who. Dungeon of Signs uses the whiskey rule, but I think I saw this somewhere else? Sorry.