Spiritually speaking on R&S is Banana simply a shorthand way of expressing the following?

Commutators Any quaternion can be written in a polar form. This is identical to Euler's formula except that the imaginary unit vector i is replaced by the normalized 3-vector. The two are equivalent if j = k = 0. Any quaternion could be the limit of the sum of an infinite number of other quaternions expressed in a polar form. I hope to show that such a quaternion mathematically behaves like the wave function of quantum mechanics, even if the notation is different. To simplify things, use a normalized quaternion, so that q* q = 1. Collect the normalized 3-vector together with I = V/(V* V)^.5. The angle s/(q* q)^.5 is a real number. Any real number can be viewed as the product of two other real numbers. This seemingly irrelevant observation lends much of the flexibility seen in quantum mechanics :-) Here is the rewrite of q. The unit vector I could also be viewed as the product of two quaternions.For classical quantum mechanics, this additional complication is unnecessary. It may be required for relativistic quantum mechanics, so this should be kept in mind. I think it would explain a lot