I am trying to work out some basic aspects in the theory of quaternions for some work in physics I'm doing. I have went through complex analysis and saw that the only way division ( and hence the derivative) could be defined was through a numerical definition of (i). My question is does there exist a notion of a quaternion derivative even without a numerical definition of ( k, quaternion part) ? I have some pieces of a possible answer, but am far away from meaningful mathematical consistency. I would greatly appreciate if someone would push me in the right direction.