The character had all but completely lost his mind when Rick found him hiding out in a small abandoned town four years ago. During their encounter, Rick asked Morgan about his son Duane who Rick had met in the very first episode of the entire series. In fact, it was Duane who clobbered Rick with a shovel outside of their house before the Atlanta-based survivors could talk all of this new world's treachery.
Morgan told Rick that he was unable to shoot his reanimated wife and his son Duane was frozen, as well. "She was just... Just on him," Morgan said.
"And I see red," Morgan went on. "I see red. Everything is red. Everything I see is red."
Now, Morgan was crazy and talking in all sorts of metaphors during the season three episode but it feels pretty clearly laid out that Morgan saw blood and Duane was dying before his eyes. However, when Rick asked Morgan, "Is he dead?" Morgan told him, "No."
He did follow up his claims of Duane not being dead with, "He will be."
The chances of Duane surviving are extremely slim even if Morgan's delusional description of the events are inaccurate. If he were to survive a separation from his father, Duane was not equipped with survival skills to make it through the treachery of the apocalypse. The only reasonable explanation for Duane surviving today is for him to have met with another group while separated from his father and finding a way to survive with them.
Morgan does, however, seem quite sure that Duane is dead now. In the wake up Benjamin's death in the Bury Me Here episode, he mistakenly described his son's death when discussing the events which killed Ben.
Maybe, just maybe, Duane is out there somewhere with Morales, hightailing it out of Birmingham only to end up at Alexandria, the Sanctuary, the Hilltop, or the Kingdom for an eventual reunion with his long, lost father.
MORE WALKING DEAD: How Benjamin's Death Is Important / How Richard's Death Launches The War / Melissa McBride Wants A Rick/Carol Reunion / Lennie James On Morgan And Rick Going Forward / Will Carol Kill Negan? / Why Carol Does Not Blame Daryl For Lying