We are running two types of fish in our system here at The Factory. Catfish and Australian Bass, these fish co-exist well and are seemingly happy after adding a few hiding spaces and blocking off the pump (I lost a few that got sucked up into the submersible pump).

The grow beds are now showing the results I had expected to see - that being that the charcoal is out performing the expanded clay in spades! I scattered lettuce seeds over both beds and put in garlic & onions (some Walking Onions and some seed from other various varieties) and now it is clear to see the difference. I added another bed later that is a mix just to see how this performs. I'm pretty convinced that the charcoal is the way to go though.
It turns out I'm not Robinson Crusoe (in fact Robinson Crusoe wasn't Robinson Crusoe either, he had Friday so wasn't alone) in using charcoal. The power of knowing how to use google (well, that belongs to Naurelle, not so much me) has shown plenty of others have discovered this as well. Either way, I did have the idea without influence and just went with my gut using it as a growing medium. It makes perfect sense, what do you use in filters in an aquarium? Charcoal. Bio-char is spread to improve yield and soil so it is an obvious step as far as I was concerned to try it as a medium in an aquaponics system - so I did, and it worked in the trial set up so in this system I chose to feature it.

We are using a submersible pump that runs on 240 volt, we have solar power that feeds into the grid but decided against a full solar set up for now. I'd liked to have investigated the option further of 100% solar run but time was of the essence so just went with this version. The pump we already had so that is why we chose that one, there are many choices and I won't bore you with this here, you can easily research it if you want and the other option would be to speak with pump supply shop staff as I'm sure they would point you in the right direction.
We run 1" PVC pipe clipped to an overhead frame (this frame also has the power conduit from the shed to the switch for the pump) and using T connections with a 90° elbow on the last one, droppers down to a tap so I can regulate the flow rate. It took a bit of fiddling to get the flow rate working and I find it can still be finicky - too much and the bed can flood, too little and the bell syphon doesn't work.
The water then returns to the fish tank, on ours we have two pipes that run directly back at the height (the grow beds are on stands) of the overflow/outlet and the other one has to run through a series of bends to allow the syphon system to operate and down under the gate. It runs into a larger 2" pipe then back up and into the fish tank. I had to mess around with this to make it work due to the position of the grow bed in relation to the gateway.
Bell syphon - this is a simple system to allow the beds to flood and drain. In my old system I did not have this, I had a permanently running flow which worked but I don't believe as well. So what the bell syphon consists of is a 100mm pipe (mine are 300mm long but it depends on your bed depth as to how long you will require it) with 2 end caps. The bottom end cap has a hole in the centre to take a 1" fitting (I obviously use tank fittings with flat washers instead of corrugated ones. They are easy to find at any plumbing supply shops. It must have a female end for the inner pipe to screw into) and small holes drilled all around it to let water into it. It also needs a series of slits to do same (I will try to get some photo's up at some stage). The top cap is just to stop debris entering. Then a 65mm pipe at 275mm with an end cap and some checked out squares on the bottom. I made them 15mm high and have played a bit with the distances between the checks to let the water in. Basically you want 3 legs for the pipe to stand on with 3 gaps to let in water. It looks like a Mercedes Benz badge. 
Finally,  a 20mm pipe with a 20 to 25mm adapter on one end and a 25 to 20mm reducer on the other (I started with the pipe at a length of 250mm overall including the reducer and adapter but shortened to match the medium depth). This is what allows the water to begin to syphon. The adapter screws into the 1" fitting (female end) which goes through the end cap of the 100mm pipe and then through the bottom of the grow bed into the return pipe to your fish tank (the male thread). The 20mm pipe stands up and the 65mm pipe sits on top of the flange on the 1" fitting and over the 20mm pipe, the slots allow water to enter. This is inside the 100mm pipe.
Sound confusing? https://www.facebook.com/slivman/videos/237325725383632 may help a bit, it is a video of the system during construction.

The fish are fed; do their thing and the nutrient rich water is pumped out into the grow beds before returning to the fish tank minus the nutrient which the plants take up. Water recycles constantly so it is a very water savvy way to garden, with the added bonus of harvesting fish to eat as well. Of course if you don't eat fish then you can either give them away or just use an ornamental species like Goldfish.

If you visit the yard I can further explain how this all works too.