(So does the hammer, but one idea per week, right?)
Percieved problem: the sword clearly outclasses the axe. (If you think it does not, please explain in the comments in what aspect is the axe better than the sword. Because the sword is better in some aspects.)
Proposed solution: change axe special attack. Instead of a shockwave, a fissure (or lot of small fissures) would appear with the same range. This deals the same 1 damage and knocks over, but it can't be blocked. (You can still jump to avoid.) Second difference, it should appear fast enough that you can't just simply walk away from its path.
Optional: the fissure could stay open for a second or two (widening and shrinking once or twice in animation.) This would increase its effectiveness in large fights, temporarily restricting the movement of enemies who don't get caught in it.
So, how would it be better? First, you have to jump to avoid, which exposes you way more than blocking. Second, easier targeting because of increased speed.
This change would make axe a strong weapon choice.
Optional: change charged attack too, to diversify melee weapons. Instead of the fancy spinning like with the sword, charged attack should be a brutal, simple strike, without moving forward, similar to the basic attack, dealing 2 damage, except that it can't be blocked.
I know, I know, unblockable attack is blasphemy, but I argue this would actually rather be a nerf. First, 1 less damage and no knock over. Second, to use it you have to walk right in front of your opponent, exposed. The enemy can roll, kick, bash, or hit you twice before you use it.
So you judge which combination of theese optional ideas creates a balanced axe. But with all of them implemented, even if it is going to turn out overall weaker than the sword, it is at least going to be different, with its own character, not simply worse in almost every aspect.
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