Doubts That Linger is an Act 2 companion quest for Aveline in Dragon Age II.
Acquisition
This quest will become available after completing Night Terrors if Aveline accompanied Hawke into the Fade and accepted the Desire Demon's offer.
Walkthrough
Travel to Viscount's Keep. Hawke can speak to Aveline, who is reflecting upon her betrayal in the Fade and apologizes. She explains that what happened was not about pining for Wesley, as she knows that he is gone; it was about seeking redemption for her failure to save him. Hawke can either comfort or clash with Aveline.
Result
Choosing the option 'Even for me?' (mage Hawke) / 'You don't mean that.' (rogue/warrior Hawke) results in 
Aveline: rivalry (+5)
Choosing the option 'Blame your memory' results in Aveline: rivalry (+10)
Choosing the option 'With Circle control, I agree' results in 
Aveline: friendship (+10)
Notes
- This quest, along with all other "Night Terrors aftermath" quests, supersedes any other companion quests, aside from the initial "Visit" quests issued at the beginning of each Act. In Aveline's case, it also comes before The Captain's Condolences and puts on hold any progress in The Long Road until the matter is addressed.