Good ol' Hank Henshaw, managing to get into both Superman and Green Lantern's Rogues' Gallery is no small feat. Suffering from a cosmic accident that left Hank cursed with immortality, and drove his wife to madness, The Cyborg is a tragic figure who elicits our sympathies to a point. And then we realize just how evil he truly is.
Hank made his first appearance in Reign of the Supermen, soon after Kal-El's death by Doomsday's hands. If Hank wanted people to really believe he was Supes resurrected…ish, then he should have came up with a better look than half Last Son and half T-800. But in the end, convincing the President that he was Supes' heir was just one step toward his end game, which climaxed with his role in the destruction of Coast City, Hal Jordan's hometown.
Downloading his consciousness into a device planted on Doomsday is just one of the reasons why Hank is still alive and plaguing Superman; he may not have Anti-Monitor epic consequences for his villainy, but he is consistent. Which means he's most likely not going anywhere. Like he said himself during the Sinestro Corps campaign against the GL corps: "Over the years, I have been detonated, burned and disintegrated. Yet I continue to survive."