Bleeding Out
A character who critically glitches a Damage Resistance test which puts him into Overflow has ruptured a major artery, and is Bleeding Out. The character takes one box of unresisted Physical damage every (Body) combat turns. An Intuition + First Aid (2) test may be performed on the character to staunch the blood flow for (Body) turns, preventing one round of blood loss. Characters with a Platelet factory or a Blood Circuit Control System may roll Body every turn in which they are bleeding out; if they achieve more hits than their current damage level, the blood flow is staunched by their 'ware and they stop Bleeding Out.

Damage Overflow and Stabilization
A character who reaches his Damage Overflow begins taking one additional box of Physical Damage every (Body) minutes until dead or stabilized. Characters who are Bleeding Out (see above) take damage every (Body) Combat Turns, instead. Stabilization requires an extended First Aid (4, One Minute) test. Stabilizing a character who is Bleeding Out adds a modifier equal to the character’'mage due to Damage Overflow.

Final Moments
Once a character has run out of Damage Overflow boxes, they are on the razor's edge between life and death. At the GM's discretion, such characters should be allowed to make a Willpower + Magic test (mundanes simply use Willpower), with a Threshold of 10 - Essence. If they fail, their soul departs from their body - never to be seen again on this plane.

Success is somewhat stranger. The final moments of death release a tremendous amount of mana. Even mundane characters find themselves capable of Astral projection and journey to the metaplanes - although this journey is absolutely final, and where precisely they go has been a subject of debate for millennia. Regardless, the connection between body and spirit is rapidly deteriorating - the body is losing 0.05 Essence per combat turn, and once its Essence is reduced to 0, death is permanent. At that moment, what happens to the soul is up to it - most move on to Elsewhere, while a few remain, effectively becoming ghosts (treat as Free Spirits of Man) with their astral body's attributes and a Force equal to their original Essence.

If medical attention manages to revive the character during this time, they are forcibly pulled back into their body, and any Essence lost while dying is permanent (barring treatments such as 'Cellular Repair' from pg. 88 of Augmentation).

What do you think, Sirs?

The first to house rules are pretty vanilla. If you think the story benefits of that level of detail are worth the book keeping go for it. I don't think that a platelet factory would help much with the scale of wound you are talking about. I don't care what your platelet count has been jacked up to; they don't do jack to help stop an bleeding artery.

The Final Moments bit though? Not at my table, all death is final. Your soul doesn't depart. It turns into mana just like your body turns into meat. Anything else is easily romanticized and not bleak enough for the genre. Loved ones and teammates don't cross the veil and float off into the cloud of unknowing or some such. They get annihilated. If you are awakened you get to see how totally the annihilation was.

Change Bleeding Out to occur on a Glitch damage resistance test that puts you into overflow. Add Bleeding Out to Critical Glitch options of Severe Wounds (Augmentation 120).

Remove the effects of Platelet Factories etc from bleeding rate. First Aid can stabilize a bleeding character as normal, if delivered in time (see below).

Change First Aid from Complex Actions to Minutes. First Aid may be rushed as Complex Actions with a -6 dice pool. Rules otherwise remain the same.

I personally do not like the Final Moments. Doesn't fit with the game.

EDIT: A character who's overflow boxes are filled is considered dead, but can still be revived in the next few minutes with medical treatment. Once the character's overflow boxes are exceeded, death is final. This is to represent how people can, in RL, be brought back from the dead if treated soon enough.

Bleeding Out is also covered under Heavy Damage as Blood Loss (Augmentation 121)

Unlike those above, I do like the final moments concept. I would have tied it to Edge, myself, rather than Willpower, thus making it more of a luck-related concept than implying you can stave off doom by gritting your teeth.