A lot of us already know what happened, but personally I thought it should have been explained in greater detail rather than with the subtle hints they gave. So, although I already knew what had happened, I've been looking for more knowledge on the matter, and so far this is the best explanation I have found. Would have been nice to know how the C2 writer would have written C3, but Stephen Hall (?) did a great job.

At the beginning of Crysis 3, it is revealed that Alcatraz was lost when the spire was sabotaged at the end of Crysis 2. The memories and consious being of Laurance Barnes embedded in the suit assimilated Prophet's mind into Alcatraz's body. But in Crysis: Legion it is shown that Alcatraz still has his mind, but has bonded with the memories and knowledge of Prophet following Crysis 2, so this part was possibly retconned.
In Prophet's character information, it is said that Barnes was no more than a ghost haunting the machine...A deadly weapon system made from alien upgrades and knotted nano-musclepacks wrapped around a stolen corpse, dreaming it was once a man named Laurance Barnes.

It was also stated in a nanosuit log that the Alcatraz personality file became corrupted beyond repair (42% of data corruption) after his mission to repurpose the Ceph black spore, and therefore it was placed in "indefinite storage after instabilities detected", meaning that Laurence Barnes became the only active personality in the host body, therefore Alcatraz's mind can be considered dead or dormant. It is unknown if his personal mind would return in absence of the Nanosuit, but it's impossible for him to be released, since his body is technically a corpse fueled by the suit's symbiotic nature and became its integral part through the nanites fusing to it.
This was also interesting:
It is implied that Alcatrz's injuries have been mitigated by, repaired by, or supplemented by the Nanosuit. If the former two are true, then it can be argued that Alcatraz's body, like Psycho's, would have a chance of survival if his suit were to be skinned by C.E.L.L. It is unknown if Alcatraz's mind would return to dominant control of his body, if he would continue to share it with Prophet's mind, if Prophet would continue dominant control, or if both would die, leaving an empty shell. It is also hinted that Alcatraz may be slowly eaten away and overwritten by Prophet's personality in the Nanosuit during Crysis 2, with Hargreave referring to him as Theseus during their meeting in Roosvelt Island, a subtle reference to Theseus' paradox.

This is the part where I nod *nods*