What's going on with James Holden's visions of Joe Miller in The Expanse, with the ghost claiming to be an investigator for the Protomolecule? Miller first appeared as a lead protagonist in The Expanse season 1, and the detective quickly became obsessed with solving the mystery of Julie Mao's disappearance. With each clue, Miller's interest in Julie evolves into a romantic infatuation, but solving the case ultimately claims Miller's life, as he boards the Protomolecule-infested Eros station to find whatever remains of Mao. When the two finally meet, Julie is unconsciously acting as the pilot of Eros as it rockets through space, but Miller is able to talk her down, forcing a (reasonably) safe crash-landing on Venus.
Miller is, naturally, presumed dead after the Eros incident but Thomas Jane's character appears to return towards the end of The Expanse season 3. After the Ring Gate forms near Uranus, a ghost of Miller starts speaking to Holden, prompting him where to go and what to do next. No other character can see Miller, leading the Rocinante crew to question their captain's sanity, but every medical test upon Holden's brain activity and well-being comes back clean. The ramblings of this vision aren't always explicit and don't always recall previous visits, often leaving Holden livid with frustration.
In the construct's own words, the Miller ghost is a "non-local quantum hologram" sent by the Protomolecule. In layman's terms, this means the figure appearing to Holden is the voice of the Protomolecule that has taken the familiar avatar of Joe Miller. Communicating with Holden remotely, these commands are the Protomolecule vocalizing its intentions. The sentient blue goop chooses this particular form because Miller became one with the Protomolecule on Eros and, as seen on multiple occasions, the substance can inherit the data and qualities of the things it absorbs. This explains why the vision seems to share certain personality traits with the original Miller, including his obsession with case-solving, as well as his memories.
Rather than Julie Mao, however, this version of Miller is working on a far more ancient mystery. Referring to itself as "The Investigator," the Protomolecule's intention is to find out what happened to its now-extinct creators. Something wiped the Protomolecule builders out, and now the blue stuff wants to uncover the truth of what happened and determine whether the destroyer race still poses a threat. To do this, the Protomolecule needs a physical pair of hands, and since it can tell that Miller held Holden in high regard, the Earthling is the unfortunate puppet being encouraged to do the Protomolecule's bidding every step of the way.
How far The Investigator can be trusted remains a mystery at present. The familiar face and voice of Miller has often belied an amoral system of ethics, with Holden's actions under the Protomolecule's direction often putting others' lives at direct risk. While The Investigator is clearly not on the side of humanity, however, it is the only lead Earth, Mars and the Belt have towards understanding the Protomolecule and discovering what benefits or threats it might pose. Holden is also painfully aware that whatever came for the Protomolecule's creators could be headed for his own solar system very soon, leading to the end of everything.
The Expanse is currently streaming on Amazon Prime Video.