Hunter monsters are enemies in The Consuming Shadow that spawn randomly (however they spawn noticeably more often after the find an occult text mission) that will attempt to attack the player. When spawned, the player will be alerted with a new objective that will say something like "GET AWAY!" rather than the usual "Return to car." objective. They can spawn in every single room one enters, except for the dungeon's entrance room. Some hunter monsters can be killed, however they don't usually stay dead, nor do they drop special items. Fleeing a room with a hunter monster incurs no sanity loss. (Unconfirmed for hunters that can be killed, such as Ghost and Fat Man.)
If you're satisfied with simply fleeing instead of continuing to search for loot and clues, one can quickly check if a hunter will appear by checking the pause menu. If the 'escape dungeon' button is unavailable, then start running. (You cannot use this option if a hunter is present in the dungeon.) This is useful, as the in-game telegraphing of the appearance of the hunter only appears after the 'Dungeon Complete!' message has faded.