> Quaternions are the same idea just in 3 dimensions (so with 3 imaginary units i j k, not just i, one per plane).

I justify quaternions to myself with the intuition from [1]. In essence quaternions represent rotations in 4D, where multiplying by a "unit" (i,j,k), rotates two distinct planes by 90 degrees. The reason introducing a single unit j doesn't work is the same reason this rotation-is-multiplication trick doesn't work in 1D (or really any odd-number of dimensions). Anyways if we call this 4th axis w and pick a simple rule like ij = k then we get some nice properties like

    - multiplying by i rotates xy + zw planes by 90 degrees
    - j rotates xz + yw
    - k rotates xw + yz
    - 1 rotates nothing

Notably this definition covers all 6 unique planes. But if we want to rotate only a single plane, we have to make up a new property, something that lets us rotate say xz by 90 and yw by -90. So we make up another rule that multiplying by a unit on the right does this, which algebraically looks like ij = -ji. This is incidentally why the rotation formulas have 1/2 everywhere, because if we want to rotate xy by 90, we multiply on the left by i/2 then on the right by -i/2.

[1] https://www.reedbeta.com/blog/why-quaternions-double-cover/