Tim

He only appears in the lower half of the cavern layer and below, and that boundary is not a fixed depth; it shifts with how your world generated. The practical marker is lava. Once you are seeing open pools of it, you are deep enough. A Remix world drops the restriction and lets him turn up anywhere underground. He is rare, so a Lifeform Analyzer earns its slot if you are hunting him deliberately.

There is a spawn trick that is easy to miss. Wear one of the gem crafted robes with nothing on your head and his rate multiplies several times over, the game noticing that you are missing half a set. It has to be a gem robe: the Mystic Robe does not count despite belonging to the same outfit. Take any Wizard Hat off before you go looking, because wearing one switches the bonus back off.

He is a Dark Caster in a hat and runs the same routine. He blinks in, winds up, throws three chaos balls that travel straight through solid ground, then blinks out. The projectiles die to any hit at all, and landing a hit on Tim himself stretches his cooldown and cancels the rest of the volley, so a fast weapon or a single minion keeps him permanently quiet. He is fragile. The real danger is whatever else is sharing the cavern with you while you chase him.

He needs solid ground to land on with clear headroom above it, no lava directly overhead, a position inside the square of terrain around you, and enough distance that he does not land on top of you. The game even guesses where you are heading before measuring that gap. If nothing qualifies he skips the teleport entirely and restarts his pattern in place, which is why he sometimes looks like he has given up in a tight tunnel.

The hat is guaranteed, and it pairs with the robe you were already wearing for a magic critical bonus, so the bait doubles as the reward. His bestiary entry also completes at a fraction of the usual kill count, which is only fair given how rarely he shows up.

He is Tim the Enchanter from Monty Python and the Holy Grail, confirmed long ago by someone who worked on the game, and the achievement for killing him quotes the same scene. The bestiary makes him the Rune Wizard's younger brother.