The brief
The A4's back end had gone choppy: rear passengers complaining about a bumpy, floaty ride, the car carrying on bouncing after bumps and feeling loose over rough roads at the rear. He brought it in. That's worn rear shock absorbers. The shock damps the spring, it stops the car bouncing after a bump and keeps the rear tyres planted on the road. As the rear shocks wear, the damping fades, so the back end floats and wallows, the ride goes choppy, and the car gets less settled over bumps and through corners. Shocks fade gradually so the front-seat driver may not notice it as much as the people in the back, but it's a real ride and stability issue, so worn ones need changing.
The diagnosis
On the lift each rear shock got a bounce-and-inspect. Both were past it, weak damping, weeping, the back end carrying on after a push instead of settling in one. The fronts were still doing their job. When the pair on an axle have gone together, you do them as a set, a fresh shock next to a tired one gives you a back end that behaves differently side to side, so the call was both rears.
The work
Both rear shock absorbers were unbolted and removed, and a new genuine Audi-spec pair fitted, the top mounts checked and renewed as needed, every fastener torqued to the manual figures. With both sides back together the rear suspension was checked over to confirm everything was sitting and bolted as it should. A road test confirmed the ride had settled, no float at the rear, the back end planted again over bumps.
The outcome
A composed ride front and back, the rear settling in one motion after a bump, stable over rough roads and through corners, and the back-seat complaints sorted. The A4 went home riding properly again. Worn shocks only get softer and they take ride quality and rear-end stability down with them, so doing the pair as a set reset the back end rather than leaving one side lagging the other.