I think what we have here is a little bit of a delayed reaction to his mother’s death. He couldn’t really stop to say goodbye to his mom when she died but as he’s dying, he catches up with himself a little bit. He doesn’t linger here but adieu is a very final goodbye and it is also a word Hamlet’s father’s ghost used. Maybe this adieu completes that circle somehow. Adieu, adieu, remember me. Wretched queen, adieu.