Halcyon is the finite, pinkish-purple resource in SAROS, which you will require in significant quantities as you upgrade your way through the Armor Matrix. The nodes it can unlock have a major impact on easing difficulty and improving survivability in each rogue-like Cycle.
This SAROS guide is all about where you can go about finding all the Halcyon you could possibly need.
Red Containers
Containers are the primary source of Halcyon on your runs, offered as one of the two choices when you open them. Even so, they are not a guaranteed choice, and you may need to grind for several Cycles to amass enough Halcyon. Whenever the opportunity does arise, Halcyon is the recommended choice, no matter what the other reward is, as it has more permanent utility.
Alphas/Enemies
Both mini-enemies and enemies have a Halcyon drop chance, but once again, this is not guaranteed. With sufficient firepower at your disposal, you can potentially attempt to farm Alphas quickly and painlessly for Halcyon.
Nightmare Gates/Strands
The seven Nightmare Strands present an additional challenge, which in turn offer you better rewards including Halcyon. The Strands primarily serve to recharge your Second Chance, one of the upgrades you will be spending Halcyon on, in the Armor Matrix. Second Chance lets you recover from a losing Cycle, and is one of the first upgrades you should aim for when the opportunity arises.
Bear in mind that the Halcyon available in a given biome has a predetermined maximum, which you can observe for yourself via the World Dial. Be sure to keep track of this, so as to avoid wasting time searching every nook and cranny of a biome where you have already exhausted the limited Halcyon supply.
Collecting just 70 Halcyon will unlock the ‘Eclipse’s Gift’ trophy/achievement. The sum total of Halcyon across the biomes is over 160, so this is easily achievable.
When spending Halcyon, be sure to prioritize reducing penalties for dying, and Second Chance. These upgrades will greatly improve the gameplay experience during Cycles.
Those are all the ways to get Halcyon quickly and efficiently in SAROS.