David
Welcome to the forum.
I can think of a no. of reasons he may have this problem.
Temperature and water quality. A stressed axolotl will develop fungus.
So, first before you do anything else:
do you have any dechlorinated water already prepared in bucket (tapwater with water ager/conditioner added)
If you do, then - prepare a salt bath in a separate plastic container (ie: ice cream or tupperware container).
* Fill the container with dechlorinated water, enough water so that your axolotl is covered completely, add 2-3 teaspoons of salt per litre or per 2 pints of water and stir so it dissolves. (table or sea salt will do for the moment if you don't have aquarium/tonic or sea salt).
* Remove your axolotl from the tank and place him in the saltbath, then cover with a lid so he doesn't take fright and jump.
* Set a timer, if you have one for 10mins (15mins max.)
* Saltbaths need to be done twice a day, three times if possible - usually weekends if you have the time - until at least a week after the fungus disappears.
Once the saltbath timer beeps/rings, I suggest placing your axolotl in the fridge while he has fungus due to the fact that he has been floating. Check that your fridge temperature is not lower than 5C, by placing a cup of water in the fridge with a thermometer, check it after 15 mins. If you don't have a thermometer, then move your tank away from the window/direct light and heat and place it in the coolest part of the house.
If you do use the fridge method: Use another container with a lid. Fill it with dechlorinated water. Once your axolotl has finished its saltbath, transfer him to the fridge container. Cover with lid and wrap with a teatowel, place him in the fridge. Empty the saltbath container, fill it up again and prepare the saltbath (write saltbath on container), lid on and into the fridge. As well you will need to place a bottle or two of dechlorinated water in the fridge, this is because you need to do complete daily waterchanges on his fridge container, and can be done while he's having his 2nd saltbath, the water needs to be the same chilled temperature.
When saltbathing and using fridge method, make sure after placing him in the saltbath, to cover with lid and put directly back into fridge so the container doesn't warm up. I always use timers to remind me when I'm saltbathing, you may be better than I, as I get distracted with other things happening round and about, ie our animals/kids/husband
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As for the fungus starting, fungus only starts when an axolotl is stressed, this may take several days or even up to a week to start, though it may appear quite suddenly to us. The fungus lets us know somethings wrong.
Part of the problem could be heat and water quality. Heat, as the tank appears to be by direct light/heat source (ie window) and the fact that summer seems to be hitting Aussie, as always earlier, than here in NZ. Your tankwater needs to be under 20C.
Water quality - you should test your water for ammonia/nitrites and nitrates in your original tank.
Moving him to the bare tank was fine. But by lowering the water, which I assume was because of floating, meant that the amount of water in the tank will warm up faster especially as it is in front of the window. This adds to his stress.