Hades - All Chthonic Companions

There are 2 categories of Keepstakes (which you can equip and level in the courtyard by interacting with the white-shaped box there) - Standard and Chthonic Companions (6 in sum).

Chthonic Companions are special Keepsakes that you can get from certain Gods/characters if you progress their story (dialogue options) enough, plus give them a certain amount of Nectar, combined with the story progression. When you unlock one, simply equip it (you can equip one Standard and 1 Chtonic at the same time).

Achilles

You need to complete his "Divided by Death" Prophecy, the one about reuniting Achilles and Patroclus. To unlock it, first give 6 Nectar to both Achilles and Patroclus and make sure you've completed "End to Torment" or "Musician and Muse" Prophecy.

Afterwards, speak with Achilles to obtain the "Divided by Death" Prophecy, which requires you to purchase the "Hero’s Sacrifice" Work Order (5 diamonds). This will allow Achilles to visit Patroclus in Elysium, and complete both of their Prophecies.

Dusa

Start with giving Dusa 6 Nectars, then work on buying 12 entries under the Lounge tab (House Contractor shop) - one must be a rug, plus "Service, Deep Cleaning", and "Service, Detailing". Speak to hear again and complete the "A Place of Revelry" prophecy.

Megaera

Give her 6 Nectars, then fight Furies (her and her sisters - first area boss') until you trigger events back at Home with Megaera.

Sisyphus

Sisyphus can be found in "!" rooms in the first areas prior to the first boss during your runs. Finish his "End of Torment" prophecy, or if we translate this - you need to help him escape from his builder punishment (poor guy). Start by giving him 6 Nectars. You also need to unlock Administrative Chamber (Buy "Administrative Privilege" Work Order when you unlock it). Talk with him every time until he introcudes his "Bouldy" companion, and then talk to his boulder at least once (yeah, you can do that).

Next step is to talk to Hades, Megaera, Nyx (they need to talk about his punishment), but you also need to speak with Nyx until he speaks about the Administrative Chamber (after you unlock it of course). You will find the document about Sisyphus' contract inside afterwards. Then you can complete the "Knave-King’s Sentance" Work Order for 4 diamonds (appears when you do the above steps). That is it.

Skelly

Give him 6 Nectars. Unlock 4 Weapon Aspects (weapon upgrades) other than the "Aspect of Zagreaus". The next time you return to him, he will ask of you to kill him with the Stygian Blade when "Aspect of Zagreus" is equipped. Do so, and the Chtonic Companion is yours.

Thanatos

Give him 6 Nectars and beat him when he appears during your runs. His appearance is random, but more or less he is appears on almost every run. Sometimes after the fight, he will talk about how strong the Zagreaus is, and that means the Chtonic Companion is yours. If you don't trigger it, you just have to repeat the process of beating him and hoping to trigger the talk.