I'd just like to add I had the same problem, 2012 A6 3.0tdi with 2 zone climate. I bought the car with 44k miles, full history and it was a very clean and well looked after car.
I soon realised the heater was virtually nonexistent. Being a mechanic myself, I rather look myself than take it back to the seller and go through hassle. But I soon found the coolant was very brown and almost sandy residue in the header tank.
I decided to pull the heater matrix out, which thankfully is actually a very easy job. It's a case of removing the glove box, remove the heater fan, undo the 2 hoses and slide the matrix out (a small plastic retaining clip has to be cut off to save a lot of work). I disconnected the matrix pipes in the engine bay, and blew the coolant out with compressed air, so that no coolant leaked into the car when removing the pipes at the matrix.
I spent some time flushing and cleaning the matrix, a ton of **** came out of it! Flushed the coolant in the engine and put it all back together. Also run some coolant flush through a few times.
A few months has passed, the heater is loads better now, and the coolant has stayed perfectly clear. However it could still be improved, so I think I'll have to replace the matrix eventually. So when I do that I think I'll flush the coolant again to make sure there's nothing to block the new matrix.
I just find it so strange how it got like that in the first place. The car has no signs of any work being done to it, still has original water pump, radiator etc so there's no reason why the coolant should have ended up by that. I wonder if it was a small bad batch of coolant from the factory? Really can't make sense of it.
This is the first post I've seen with someone having the same symptoms as I did. So I just thought I would add to it in case anyone is searching for the same fault like I was a while back!