When I change my fish water I get a reading of .25 of ammonia right after I use the dechlorinator. Our town uses chloramine in the water to purify it. In my fully cycled tanks the ammonia is gone within 24 hours or less.
Occasionally the automatic chloramine injector that the town uses on its water tanks misfires and so instead of getting the correct amount of ammonia and chlorine injected they add the wrong proportions and all hell breaks loose in peoples tanks if they have the misfortune to clean the tanks on those days.
I tend to use seachem Prime as a water conditioner and usually double dose it especially in the spring and fall when the water company seems to mess around with its chloramine . You will get a false positive reading for ammonia at that time in the sense that yes there is ammonium ( not ammon ia) in the water but it is in a non toxic form because of the effect of the PRIME.
The biofilter then can eat up the ammonium.
If you have soft water the ammonia is in a less toxic form( ammonium) and will not be as harmful to your fish.
I hope you can understand this.
Ask your water company what the ph is as well as hardness and if they are using chloramine.
Some towns are irradiating the water and not using chemicals.
There is plenty to learn but your idea about the carbon and filters is good but honestly I can't think why you have that ammonia level and why it isn't disappearing.
Are you constantly adding clean water? cleaning the tank every couple of days?
Is there anything that could be decaying in the tank? sludge on gravel, dead algae on the glass? If not, its likely its coming in your water. StressCoat+ sounds like Prime in that it should have "detoxified" the ammonia whether or not you can see it on the test. Hopefully soon you will see it vanish as the ammonia gets eaten up.
Is there anything adding color to your water that could fool the test? Rust, driftwood, peat? Verify your test kit by testing something with no ammonia (maybe bottled water). Test the tap water right after dechlor to see if that is where the ammonia is coming from. 0.5 will cause stress if it isn't 'detoxified' but isn't immediately fatal.
The old Penguin 100 and a 10% water change got it down to 0.25 yesterday...I'm really thinking the established filter started knocking out the ammonia. I added three small platys on Tuesday, since I figured they were about, if a little less than, the bioload of a messy, 2-inch convict (which is what the filter was handling).
Today's my last day of Stablity...have yet to do my fish chores today (PWC, tests, feeding, etc). Hopefully the ammonia will be even less.
I'll definitely test all the water sources I've been using (with and without dechlorinator) later today and get back to ya'll.