I was trying to prove $ij = k$ from $i^2 = j^2 = k^2 = -1$ by doing:
(1) $i^2 = j^2$
(2) $\Rightarrow i^2 j^2 = 1$ (multiplying both sides with $j^2$ ($j^4 = 1$))
(3) $\Rightarrow i^2 j^2 = k^4$
(4) $\Rightarrow (ij)^2 = k^4$
(5) $\Rightarrow ij = k^2$
Obviously this is not right, especially because there is no way I could have proven it without assuming ijk=-1 according to this
Still though, I'd like to know where my mistake is, I assume it's between (3) and (4) but why does $x^{u}y^{u} = (xy)^{u}$ not hold in $\mathbb H$ ?
The mistakes are (4) and (5).
(4) because $x^ny^n=(xy)^n$ if $x$ and $y$ commute. If they don't commute, it may not hold as in your example. Indeed, $i^2j^2=1$ but $(ij)^2=k^2=-1$.
(5) because even in a more familiar structure like $\mathbb{R}$, you have $(-4)^2=2^4$ but $-4\neq 2^2$.