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Intuition is a personal thing, but here are some thoughts that might be useful. (Rigorous justifications for most claims are absent, to keep this post from getting too long. I intend to restrict myself merely to pointing suggestively and waggling my eyebrows.)

Don't think of the quaternion trick as taking 3-D space and extending it into a fourth dimension. Instead, think of the quaternions as a pre-existing four-dimensional algebra, and consider the three-dimensional subset with no real component. $i$, $j$, and $k$ are not *quite* interchangeable; you can give them an even permutation without changing the algebra, but an odd permutation does change the algebra. Let $P$ be a linear function on the quaternions that takes $i$ to $j$, $j$ to $i$, and $1$ and $k$ to themselves. Notice that $ijk = -1$, but:

$$P(ijk) = P(i)P(j)P(k) = jik = 1 \neq P(-1)$$

This fact is also true of rotations in 3-D space: you can give $x$, $y$, and $z$ even permutations with a rotation, but not odd ones.

Even permutations are a special case; more generally, you can replace $i$, $j$, and $k$ with certain linear combinations of themselves without changing the algebra—as long as those new basis elements obey the same multiplication laws (which amount to ensuring that the elements, when interpreted as vectors, are orthogonal, unit length, and have the same relative orientation). And of course, the same holds for 3-D rotations again.

The formal fact behind this intuition building is that the automorphism group of the quaternions is isomorphic to the rotation group of 3-D space, and that's why the quaternion trick works. Rotations represented by quaternions are just automorphisms of quaternions, and automorphisms must preserve the real subset. Conjugating one quaternion by another also preserves the real component of the inner one:

$$q(a + bi + cj + dk)q^{-1} = qaq^{-1} + \ldots = aqq^{-1} + \ldots = a + \ldots$$

so it's natural to represent the action of a quaternion automorphism via a conjugation. I find this to be a more helpful framing than transforming a vector into a fourth dimension and then back into three.

In order to use this trick, you can't just take any real vector space, slap an extra dimension on it, and start doing algebra. The algebra you use needs to be a normed division algebra, and those are in very short supply! By Hurwitz's theorem, there are only four: the reals themselves, complex numbers, quaternions, and octonions. The reals can't be used to represent rotations in this way because their automorphism group is trivial. The complex numbers can be used to represent a single reflection, or ‘rotations’ of a 1-D space if you like—but who would do that instead of just keeping a two-state bit? And the octonions have an automorphism group (G2) that is isomorphic to only a *subset* of the rotations of 7-D space, so they won't even do for that purpose. The quaternions really are special!