The story takes place not long after the trilogy. Neo and Trinity are dead. The machines are helping to rebuild Zion now that they have an agreement. Still, many citizens of Zion are incredibly restless and paranoid. They do not trust this "peace". These people try to recruit Morpheus to legitimize their perspective, but he fully believes that Neo saved them and will not under any circumstance disrespect his sacrifice. He can't even bring himself to entertain the idea.
This peace (stalemate) continues even while whispers of Neo sightings start echoing through the matrix. There are lots of Neo posers. He becomes, increasingly over time, a mythical "jesus" like figure who is celebrated for misunderstood reasons. Morpheus hates that Neo's sacrifice and memory are being bastardized by false narratives, and he wants nothing to do with any of it -- until he himself see a dead ringer for Neo somewhere in the matrix. Just a glimpse. But it's enough.
Now he's searching. He's starting to believe it. He talks to the oracle. The oracle tells him that many programs in the matrix are based on, or copies of real people who have lived -- what better way to make a program blend in?
Is this a program mimicking Neo and using the shell of Neo? Is this a copy of Neo? If it's a copy of Neo, is it Neo?
These are the agonizing questions at the heart of this story. And we won't know the answer until toward the end.
It turns out, the machines did copy -- and then manipulated -- Neo's code. He's in there, but he's scrambled. Confused. His memories are all wrong. His feelings are insane. He has no grounding. He has no purpose. He doesn't know who he is/was.
What the machines don't understand is that the Sati, under guidance of the oracle, sprinkled Neo's code and full memory into the matrix itself when she redesigned it. They knew the machines would try to use the legend of Neo to control humans, and they had a failsafe for such an occasion.
The longer this Neo program is in the matrix, the more it absorbs proper "Neo" code from the environment. But, for a long time, it confuses him and makes him behave erratically. Sightings of him become increasingly creepy/bizarre. He pops in and out of existence like a ghost. He's caught on CCTV doing superhuman things at random. It becomes a problem for the machines, as his existence was supposed to subtly pull tribal thinking humans toward their agenda, but his insane behavior has turned him into a massive liability. So they decide to try to shut him down.
By now, Morpheus has joined forces with every living/waking human who didn't trust this "peace" from the beginning. Now he's pissed off. He feels like they took his good friend's soul and weaponized it against humanity. He hates the machines with a deeper passion than ever. He knows that this glitchy, unpredictable copy of Neo is dangerous, but he doesn't care. He absolutely MUST find him and talk to him. He does find him -- because he knows him.
I've thought of a dozen different ways this interaction could go. In the end, Morpheus appeals to a part of Neo that seemed unlikely. Neo's pieces all fall together and he remembers who he was and what he was doing up to the very moment he died. It's right then -- at the moment Neo "wakes up" -- the agents attack. They totally obliterate Neo, and he's nothing but code/ash/nothingness. Morpheus screams "Neo!!!!" and goes charging after the agents on an obvious suicide mission. But riiiiiight when he's about to get slaughtered, time freezes. The scene reverses. Morpheus is watching everything happen backwards. He watches as the agents are about to kill Neo, and instead of that happening, Neo punches the floor, and the agents all explode.
Or whatever. I'm having fun with the end.
Before the credits roll, though, it's clear that Neo is physically dead. But his soul lives on in the matrix. He's still there. He just doesn't have a meat body.