Cole's Amulet bugged?

It does sound like a bug. First time I played, before any patches, I had the problem of an invisible Cassandra when I was at Skyhold but if I did a quick save and reloaded when standing where she should be, she reappeared. I only discovered this a good way into the game and nearly missed the Seeker quest as a result. Have you got a save just before you gave him the amulet as it sounds like you'd have to try again from there.

Other weird bugs I have encountered have been:

Headless soldiers practicing at Haven whilst I am trying to have a conversation with Cullen.
Returning from the Undercroft to find Skyhold devoid of all furniture and people.
In the final fight in Trespasser, Dorian being knocked out and replaced by his simulacrum, only for him to be sucked into the void when I used the anchor.
In the same sequence, whilst running around panicking at Dorian's destruction, noticing that Cassandra was now headless.

However, all bar the headless soldiers was simply remedied by loading the last save.

Yeah you have to have a good amount of approval from him first. Been there done that. Go to Emprise with Cole and liberate the slaves in the cages, lots of approval.

This is the bit that puzzles me. "Look for Cole" is something you should be doing as soon as you get to Skyhold. This is in the courtyard. Then he transfers to the top floor of the inn and I seem to recall you get another "Look for Cole" when you go to there. After that he is always lurking in the inn, although I occasionally had a bug when he didn't seem to be there but reloaded like I did with Cassandra and he appeared. The cut scene with him helping the woman has nothing to do with the amulet quest.

The cut scene that ought to trigger on giving him the amulet is the one with Solas and then after that the one in Redcliff with Varric. So if giving him the amulet is triggering a different cut scene, something is definitely wrong. May be you are giving him the amulet too soon before certain other cut scenes with him have been triggered. This may have to do with approval or simply not talking to him enough to trigger them.

I'd also mention that generally I always do Wicked Eyes before Into the Abyss, so may be you do have to have completed both quests and spoken to him afterwards to get all the scenes to trigger in the right order.

That is useful to know should I play the game again and do things differently. That blip that can occur with quests being connected with approval and cut scenes is something that has cropped up from time to time. The most obvious one that springs to mind was when initially they had the flags wrong on Dorian so his personal quest, that should have been triggering with approval above a certain level, was instead only triggering if approval was below the required threshold. They fixed that in a subsequent patch but it is possible that this blip with Cole should not be there but there weren't enough instances of people being inconvenienced by it for them to realise it needed fixing.