Student or Learner
Does it make sense if 'take up' goes with 'complaints'?
Yes. A shop assistant tells a customer that her complaint will be acknowledged and dealt with:
"I'll take up your complaint with the store manager, and I'm sure he'll be in touch to rectify this as soon as possible."
You can also 'take up an issue' with someone, which means confronting someone over a problem in a relationship or the workplace.
It means that your are pursuing some matter further.
(At the same time, you could say that 'we will take up this matter later when I've had time to...")
You don't 'take up' (medical) complaints with a doctor...or anybody else. You just 'discuss' it.