In another question, it was mentioned that an aquarium filter like this would help in an aquaponics setup (picture). However, I thought the rocks played the role of the filter. Am I missing something?
Based on the picture from the last question, you wouldn't need an additional filter but it would be helpful.
The rocks near the output of your pump will serve as a good breeding ground for Nitrosomonas and Nitrobacter bacteria, which are responsible for converting ammonia and nitrate (respectively, Nitrosomonas first converting ammonia into nitrite) into nitrate. Although plants can absorb ammonia and nitrite, they can't do it well enough to survive off of them and absorb nitrate at an exponentially higher rate. Also, fish can handle high levels of nitrate, but are vulnerable to even low doses of ammonia and nitrite.
However, if your pump were to fail, those colony of bacteria would quickly die due to drying out, as they need to be constantly moist. An aquarium filter, on the other hand, provides a better environment (the mesh can sustain more colonies) and even if the filter were to fail, it still remains full of water and can keep your colonies alive a lot longer before drying out.
Additionally, the filter will reduce uneaten food from getting stuck in the rocks slowing down harmful bacteria and algae growth. Finally, the movement from the filter will help oxygenate your water even further (although there is already a good amount of water movement in this system, but more is never worse).
I have an outdoor pond which is not set up for aquaponics. There are similarities as there are plants and fish in an outdoor area.
I started using volcanic rock in a layer between landscape fabric to give a home to beneficial bacteria. However with fish, plants and rainwater being added to top it up the algae just got out of control. I added a filter box like this which increased the volume where bacteria could live by a huge amount and this really helped even out the swings in nitrate/nitrite/ammonium levels.