Derivative of a quaternion-valued function over quaternions?

Question

This may be naive, but ever since I was exposed to quaternions, I wondered if it possible to introduce a notion of derivative of a quaternion-valued function over the quaternions. After all, we have a definition of distance over the quaternions, with which we can define a notion of limit with the usual $\epsilon$, $\delta$ definition, and since the quaternions form a division algebra, we can define a derivative as the limit of the ratio of the function increment over the argument increment. Is there such a thing? What properties of the standard derivative over real- or complex-valued functions carry over? Is there an equivalent to analytic functions? Would the non-commutativity of the algebra throw a wrench in the whole thing?