If you check some of the resources in the earlier question you'll find that the most useful way quaternions act as rotations is by conjugation.
Think of the $i,j,k$ vectors as orthonormal vectors in 3-dimensional space, as we usually do in physics. Every point in 3-space then is just a linear combination of these three vectors. These are the "pure quaternions" whose real parts are 0.
Given a quaternion with norm 1, call it $u$, you can rotate a pure quaternions $v$ by conjugating: $v\mapsto uvu^{-1}$. Let $w$ be another quaternion with norm 1. Then as you observed, you can rotate by $u$ and $w$ in two different orders:
$$wuvu^{-1}w^{-1}=(wu)v(wu)^{-1}$$
or
$$uwvw^{-1}u^{-1}=(uw)v(uw)^{-1}$$
which potentially can be different.
Let's try it with a few very simple choices of $u$ and $w$. Try $u=i$ and $w=j$, and see what happens to the $i,j,k$ vectors under those rotations. If we try this with $u=i$, you can check that $$i\mapsto iii^{-1}=i$$
$$j\mapsto iji^{-1}=-j$$ $$k\mapsto iki^{-1}=-k$$.
Visualize what has happened to the original triad $i,j,k$ after rotation. I'll leave the other example to you.
To customize length 1 quaternions that rotate things the way you want to, you'll have to take a look at the wiki article. Basically the idea is this: every rotation in 3-space is specified by an axis of rotation and the angle you rotate about that axis. To find your customized $u$, you first compute a unit quaternion $h$ which is normal to the plane of rotation, and then an expression like $u=\cos(\theta/2)+h\sin(\theta/2)$ turns out to be what you want. (I haven't been careful about specifying the direction and rotation or signs in this sketch, so take care when following the detailed explanation.)
