It's incredibly ambiguous, there's just as much reason for both being true and that of course is deliberate.
Reasons for him being alive:
- He would be a good character to have in Inquisition for his fate to be left to the Inquisitor.
- No body, Cole usually leaves them.
- The spirit taking control of Cole, not the 'Cole' we were shown*, could have different intentions.
- The book also made it seem like Cole killed the Tranquil when he didn't, this plot device could be used again (see first point)
Reasons for being dead:
- He was a direct threat to Rhys. Cole, or at least the spirit helping him would have eliminated this threat if his intentions were consistent.
- No body. The spirit itself, not the manifestation of Cole's memories, was talking to Lambert, maybe this spirit would hide a body.
- He has been gone for a while with no explanation.
- The 'I want you to look into my eyes' is exactly what Cole said all the other times before he killed someone.
*We still don't know what or who Cole is, whether the spirit was in control the whole time using the dead boy's memories to create a persona or whether like Wynne, he's still Cole and is just hosting a benevolent spirit that can sometimes take control. I believe this could have a huge affect on Lambert's status.