Quaternions (when viewed as vectors) are not linear functions, but the arguments to linear functions. You can add them: (a + bi + cj + dk) + (a' + b'i + c'j + d'k) = (a + a') + (b + b')i + (c + c')j + (d + d')k, and multiply them by a scalar: lambda * (a + bi + cj + dk)= (lambda * a) + (lambda * b)i + (lambda * c)j + (lambda * d)k. An example of a linear function on quaternions is the zero function. After all, zero(q + q') = 0 = 0 + 0 = zero(q) + zero(q'), and zero(lambda * q) = 0 = lambda * 0 = lambda * zero(q).
Matrices and quaternions take different approaches to describing rotations: a matrix sees a rotation as a linear function, and quaternions see rotations as a group (confusingly represented with matrices, this field is called representation theory if you want to know more).
So the answer to your question: there are linear functions that quaternions cannot describe. And quaternions can only describe a very specific class of linear functions (with some rather complicated maths behind them).
A matrix is a representation of a linear function (e.g. a function that plays nice with + and scalar multiplication). A specific subset can be used to describe rotations in 3D space. Quaternions can (arguably) do this better. But quaternions cannot be used to describe any linear function. So I do not think this makes sense for LLMs.
I think it depends more on the ratio between access time and how often you use the data. Adding two arrays that fit in L1 is already limited by access time. On Zen3, we can add two 32-byte vectors per cycle, but only store one of these per cycle. For matrix multiplication, we can do the two additions per cycle (or really c <- a * b + c) because we have to do multiple operations once we have loaded the data into registers.
I can see it be useful for data sets of a few dozen MBs as well.
Matrices and quaternions take different approaches to describing rotations: a matrix sees a rotation as a linear function, and quaternions see rotations as a group (confusingly represented with matrices, this field is called representation theory if you want to know more).
So the answer to your question: there are linear functions that quaternions cannot describe. And quaternions can only describe a very specific class of linear functions (with some rather complicated maths behind them).