But, I wonder if you math folks know of a definition of derivative of a quarternionic function with respect to quarternionic variable, generalizing the Cauchy-Riemann definition [1] of complex differentiation?

[1] https://en.m.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equat...

B) I think we should pass a hacker news law via referendum (that's a thing, right? It should be!) that any article mentioning quaternions must also generalize to Octonions, or at least gesture in that vague direction to pay respect. As the first paper I found on Octonion Differentiation says best:

  Each Cayley-Dickson algebra Ar+1 is obtained from the preceding Ar with the help of the so called doubling procedure [1, 14, 17]. This gives the family of embedded algebras: Ar ↪→ Ar+1 ↪→ .... For a unification of notation it is convenient to put: A0 = R for the real field, A1 = C for the complex field, A2 = H denotes the quaternion skew field, A3 = O is the octonion algebra, A4 denotes the sedenion algebra. 

  The quaternion skew field is associative, but non-commutative. The octonion algebra is the alternative division algebra with the multiplicative norm. The sedenion algebra and Cayley-Dickson algebras of higher order r ≥ 4 are not division algebras and have not any non-trivial multiplicative norm. Each equation of the form ax = b with non-zero octonion a and any octonion b can be resolved in the octonion algebra: x = a−1b, but it may be non-resolvable in Cayley-Dickson algebras of higher order r ≥ 4 because of divisors of zero.

  Therefore, in this article differential equations are considered [only] with octonion or quaternion variables for octonion or quaternion valued functions.