This enchanted axe has a somewhat conventional appearance, being more akin to a woodcutters axe then a battleaxe.
To the wielder, all magic becomes visible to some degree, visualized as plants and trees - although with features hinting at their nature. This tends to follow the cultural expectations - humans would generally see beneficial effects as beautiful flowers, while black magic would appear more like thorn bushes, ugly giant mushrooms, etc.
Only long duration or permanent magic is seen this way - magic which has come and gone is not visible. Beings with magical effects or with magical natures will appear enshrouded by plant-life - vines, tendrils, leaves.
An effect will also have a visible tie to its caster or creator - roots which disappear into the ground, vines which extend up and out of view, etc. These can be targeted by the wielder as well, and cutting them could interrupt a spell or remove the caster’s control over it.
This plant life can be attacked with the axe, damaged and destroyed, though this is not without risk. Spells provide unique dangers to the bearer of the axe, and may be able to strike back.
The more powerful the spell, the more likely it will fight.