Alcatraz Pocket Dimension

"Do not speak so lightly of hell! Relax, Tak. I didn't mean the real thing."

- Takeo Masaki and Tank Dempsey in the Blood of the Dead intro.

Purgatory, also known as Hell was a realm seen in the Aether storyline of Treyarch's Call of Duty: Zombies series. It is a realm residing in the Afterlife, and, according to Edward Richtofen, is used to torment the souls of "both the damned and the doomed". It was presumably closed-off after the events of Blood of the Dead.

Purgatory has only been known to take the form of Alcatraz Island, and the prison that reisdes on it.

It is inhabited by the Warden of Alcatraz, tasked by the Shadowman to torture souls.

Overview
Purgatory was created by the Shadowman as a means to trap Primis, and to gather Edward Richtofen's blood, as to use it to open a portal to Agartha. He also had the souls of 4 mobsters (Albert Arlington, Salvatore DeLuca, Billy Handsome, and Finn O'Leary) imprisoned in Purgatory for unknown reasons. In order to reach Purgatory,

While trapped in Purgatory, the 4 mobsters attempted to build the plane nicknamed "Icarus" by Arlington, otherwise known as the "Weasel". When flying on Icarus, it is struck down from the sky, causing the 4 mobsters to land on the Golden Gate Bridge. Soon after, the Warden reveals to the group the truth of what occurred to them. In Dimension 63, a failed attempt to escape the island resulted in the other 3 mobsters murdering Al on the rooftop. After realizing this, the mobsters, in Afterlife form, return to the Golden Gate Bridge to finish what was started so long ago.

Good Ending (Canon)= If Arlington kills the other 3 mobsters, the Cycle ends, allowing the 3 to roam freely around the island. Bad Ending= If the other 3 mobsters kill Arlington, the cycle continues and the events of Mob of the Dead are repeated. After defeating the other 3 mobsters, Arlington takes the nickname of Icarus and becomes a spirit in the form of a bird.

Shortly after, the group known as Primis arrives in Purgatory to gather blood vials. Deviating from "the cycle" that the Kronorium lays out, Primis is trapped in Purgatory, with seemingly no way out. However, this is soon found out to not be the case.

At some point, after a cycle, Great-War Richtofen is put in a cryopod, alongside a version of Victis.

While trapped in Purgatory, Primis discover the house of the Warden, where his decaying corpse is found. They attempt to retrieve the Kronorium, but Arlington appears and steals it from them. They then must complete challenges presented by the other spirits of Alcatraz. They return to the Warden's corpse, where they are captured by the Warden and locked in cells. It is soon revealed that the Warden plans to use their blood to open a gateway to Agartha for the Apothicons. Unfortunately for the Warden, Arlington returns, giving Primis their equipment and weaponry back, and helping them sneak past the Warden. When the Warden discovers Primis has escaped, Arlington and the spirits of his fellow mobsters (notably Sal and Finn) hold the Warden back to allow Primis to get to the lab. When there, they begin to defrost the cryopod containing a Richtofen who has already experienced the events of the cycle. After defeating the Warden in battle, and Richtofen entering the Dark Mechanism, the spirits of Alcatraz are freed from the realm. Great-War Richtofen hands Nikolai the Kronorium, leading them out of Purgatory. The previous Richtofen, stuck in the Dark Mechanism, has the blood drained from him and dies.

After this, the dimension was closed-off, and presumably destroyed.