Can you engage an enemy that is already engaged with another investigator?

Yes, as long both investigators are at the same location you can spend an action to engage a enemy that is already engaged with another investigator.

The enemy is removed from the threat area of one investigator and placed into the threat area of the other investigator.

Cool. That makes things just a little bit easier.

Now when an enemy engages during the enemy phase it still attacks, right?

10 minutes ago, Purple Orc said:

Cool. That makes things just a little bit easier.

Now when an enemy engages during the enemy phase it still attacks, right?

Depends. Hunter enemies do, yes, since that keyword resolves before the "enemy attacks" step of the phase. But an effect that cause an enemy to engage at the end of the phase wouldn't since that step has already resolved.

Okay so I just realized we were playing it wrong.

We were having the non-hunter enemies engage during the enemy phase and then attack.

Now I realize they engage during the upkeep phase.

Thanks for the help

I think you are making it more complex than it needs to be. Enemies engage whenever they have an opportunity to do so, which includes when they are first played during the mythos phase, when a hunter enemy moves to an investigators location and when an investigator moves to a readied enemy's location.

There are a lot of sub rules, but basically if an enemy is not exhausted, not engaged with another investigator, not aloof and in the same location as an investigator, it engages immediately no matter what phase it is.