Spoilers ahead for The OA Season 2. The OA is really taking Homer's name literally, because now, he's had to embark on a long Odyssey to find his way back to Prairie. When the show picks up, we learn that everyone in Hap's group of captives — including Hap himself — has successfully made it to a new dimension. And somehow, they all ended up in the same one. But why didn't Homer jump to this new OA dimension, too? Is he even still alive? By the end of the season, we do eventually get some answers.
As Hap reveals to Prairie, he took Scott, Rachel, Renata, and Homer into a field and had them do the movements. Homer initially tried to run away, but changed his mind. "We'll get him on the other side," Rachel said, referring to Hap. In Season 2, Episode 2 we see this from Hap's perspective. When Hap woke up in the new dimension, he was leading a group therapy session with the others as his patients. Scott immediately attacked him, but he was stopped by a different version of Homer. This dimension's Homer is a third year resident named Dr. Roberts working under Hap. They're cohorts, and maybe even friends. He trusts Hap implicitly.
However, Prairie recognizes this dimension from Homer's near death experience — what he saw when Hap temporarily drowned him. Certain elements of the Treasure Island facility line up with what he saw, like someone trying to grab him while he was hiding in the ceiling, and him trying to eat a sea anemone. The NDEs do seem to act as glimpses into other dimensions, so perhaps Homer was caught between the two after they jumped, or Dr. Roberts was for some reason suppressing him, as Prairie did with Nina Azarova.
The latter seems to be the most likely case. After Prairie figures out how to "unlock" Nina in Episode 8, she goes to Treasure Island to help Homer do the same. She repeats gestures from his life in the other dimension, like the kiss they shared and the way they'd press their hands together against the glass in Hap's cage. It's not clear why Homer was being suppressed, while the others immediately overtook their alter-egos, but he does eventually remember who he is (or was? This show is so confusing). When Renata asks him if he's Dr. Roberts or Homer, he replies "both."
After Homer finally realizes who Hap is, he runs to punch him just as Hap is trying to transport OA to another dimension against her will. Hap shoots him, but OA isn't concerned: she says they'll find each other in the next dimension, where that iteration of Homer is still alive. The jump is successful and though OA is injured, she'll likely be OK. Otherwise, the show would kind of be over...right?
We don't get to see Homer in this dimension, but that tees up Season 3 perfectly. The next chapter (assuming Netflix renews OA) will likely be about OA and Homer's journey to find one another — as well as their continuous battle to take down Hap.