Character death in D&D 5e typically unfolds like this: You drop to 0 HP, fall unconscious, lie on the ground for a few rounds, fail a few death saving throws, and within 2-5 rounds, your character dies.
I propose an alternative to make death more engaging and give players additional choices. When your character drops to 0 HP, you can choose from the following options:
Risk it all and fight on!
You remain at 0 HP but do not fall unconscious. If you take damage, it counts as a failed death saving throw. Each round, you make death saving throws as normal, but with disadvantage. However, if you roll a natural 20 on either die, you regain 1 HP.
Cling to life and keep moving.
You drop to 0 HP but can still communicate and move at half your speed. Death saving throws are made as normal, but rolling a natural 20 does not restore 1 HP.
Hope for the best.
Use the standard rules for death saving throws and unconsciousness.