I was thinking about what's possessing Max yesterday, and somewhere along this tangential train of thought, my brain was like, "What if Max was possessed by the ghost of Davey Jones?"
First off, I was thinking about how Max might be possessed by a ghost, not a spirit. So far, we have no example in the comic for if ghosts can posses people, but as far as I know, there's no reason they wouldn't be able to except for the fact that if a spectral is using a spirit it could eat the ghost. Therefore, a ghost that could posses someone would need to be powerful. Secondly, Boss Leader can see the forms of spirits with mental links to people within their room in the dreamscape. She was unable to find anything else besides Scrapdragon in Max's room, but she did look based on "instincts." Maybe ghosts interact differently with her power, allowing them to stay hidden. Max also never sees anything else when pulled into a spirit trance with Scrapdragon, despite the fact that something else is obviously possessing him. Perhaps ghosts don't show up in "spirit" trances either. We know that multiple spirits can show up in the same spirit trance because of Mr. Spender. Thirdly, for the ghost theory, we know that spirits can't pass in and out of the barrier for Mayview, but ghosts can. With the timing of Max beginning to see shades, it is possible that he was possessed before entering Mayview. It is still unclear whether Forge could use Johnny to pass through the barrier because he was only partway through, or if KOed spirits in some way don't "count" if they possess someone who isn't a Spectral yet. If ghosts can pass through the barrier anyway, a ghost might have a different motive for possessing someone. Or, perhaps, ghosts of Spectrals still can't pass through the barrier.
Why Davey Jones?
He is a vampire, a "corrupted human," who "used to be a Spectral," but is now functionally dead. Spectrals are humans who have produced a ghost before they died. Therefore, becoming "undead" would probably separate that ghost from the human, splitting them in two. The thing possessing Max was described as a "broken god." Davey Jones has been shown to be powerful, and in power. He's plotting something that probably isn't good, and seems to have beef with something also implied to be a powerful entity, the "Witch." He is of the opinion that the entire town of Mayview is "his." Another thing: his energy color is greyish. Is that a diluted form of black, his possible spectral energy color, now that his ghost is unattached to his body? If so, that would match Max's color, and since you can become a spectral from "exposure to the supernatural," Davey's ghost could provide the source for Max's spectral energy color by possessing him. Something that could possibly be important: Davey Jones definitely has a sense of humor, something the thing possessing Max said he "didn't get [in whatever split that occurred to make it "broken"].
"Davy Jones" is the name of a pirate legend, you might know from Pirates of the Caribbean. One source claims that Davy Jones "perches in the rigging on the eve of a hurricane." We know that some catastrophic event occurred in Mayview in the past called the Maelstrom, and that something catastrophic is going to happen again "by the summer's end." Is Davey Jones involved in one or both of these events? The Doorman recognizes the thing possessing Max, and mentions it has "returned," so it was definitely involved in something in the past.
I may be way off base, but I just wanted to share my thoughts. Feel free to correct something I got wrong, or post a counterpoint to my argument.