Purgatory = Hades?

"The Catechism of the Catholic Church defines purgatory as a “purification, so as to achieve the holiness necessary to enter the joy of heaven,” which is experienced by those “who die in God’s grace and friendship, but still imperfectly purified” (CCC 1030). It notes that “this final purification of the elect . . . is entirely different from the punishment of the damned” (CCC 1031).

The purification is necessary because, as Scripture teaches, nothing unclean will enter the presence of God in heaven (Rev. 21:27) and, while we may die with our mortal sins forgiven, there can still be many impurities in us, specifically venial sins and the temporal punishment due to sins already forgiven."
https://www.catholic.com/tract/purgatory

"Hades is the state in which all the dead exist. In the New Testament a descent to Hades may simply refer to someone's death and disembodied existence. In this sense even Jesus enters Hades."
https://www.biblestudytools.com/dictionary/hades/#google_vignette

"The Rich Man and Lazarus
19 “There was a rich man who was dressed in purple and fine linen and lived in luxury every day. 20 At his gate was laid a beggar named Lazarus, covered with sores 21 and longing to eat what fell from the rich man’s table. Even the dogs came and licked his sores.

22 “The time came when the beggar died and the angels carried him to Abraham’s side. The rich man also died and was buried. 23 In Hades, where he was in torment, he looked up and saw Abraham far away, with Lazarus by his side. 24 So he called to him, ‘Father Abraham, have pity on me and send Lazarus to dip the tip of his finger in water and cool my tongue, because I am in agony in this fire.’

25 “But Abraham replied, ‘Son, remember that in your lifetime you received your good things, while Lazarus received bad things, but now he is comforted here and you are in agony."
https://www.biblegateway.com/passage/?search=Luke%2016:19-31&version=NIV

"As Jesus neared death, He said, “It is finished” (John 19:30). His suffering in our place was completed. His soul/spirit went to hades (the place of the dead)."
https://www.gotquestions.org/did-Jesus-go-to-hell.html

"Hades is the Ancient Greek god of the Underworld, the place where human souls go after death. In time, his name became synonymous with his realm. It has to be said unsurprisingly – since he barely left it. Appropriately, the most significant myth related to Hades concerns one of the very few times he did – to abduct Demeter’s daughter, Persephone."
https://www.greekmythology.com/Olympians/Hades/hades.html

"And the sea gave up the dead who were in it, Death and Hades gave up the dead who were in them, and they were judged, each one of them, according to what they had done."
https://www.bibleref.com/Revelation/20/Revelation-20-13.html

"The Book of Life is a record written by God before the creation of the world, listing people who will live forever in the kingdom of heaven. The term appears in both the Old Testament and New Testament."
https://www.learnreligions.com/book-of-life-700739