The 4 real variables a0, a1, a2, a3 can be represented by functions using the conjugates as a basis. That changes with the following example, a tricky approach to the same result. Because the 3-vector goes to zero faster than the scalar for the differential element, after the first limit process, the remaining differential is a scalar so it commutes with any quaternion. This is what is required to dance around the e1 and lead to the cancellation. The initial hypothesis was that complex analysis should be a self-evident subset of quaternion analysis.
Quaternions have three imaginary axes, which creates the factor of three.
The derivative of a quaternion applies equally well to polynomials. This is the expected result for this polynomial. It would be straightforward to show that all polynomials gave the expected results. This is actually consistent with principles of special relativity. For timelike separated events, right and left depend on the inertial reference frame, so a timelike derivative should not depend on the direction of the 3-vector.