In my spec v I use seachem matrix as the biomedia, and seachem purigen for chemical filtration. I tossed the carbon.
Carbon can be useful, along with water changes, if you need to remove chemicals after treating your tank for something, but in general, I don't think it's necessary.
I use purigen because I have driftwood in my water which can cause a yellowish tinge, and I also put in dried leaves from time to time, and since I don't boil the tannins out first, those can definitely tint your water. The purigen soaks all those up and keeps the water crystal clear.
In a spec v, you can put a bag of purigen in the space between/under the two chambers very easily. I don't know that you need it though. Your tank looks very lush and healthy.