Can quaternions be useful for integrals? Lets assume we want to find a closed form for $\int_0^1 f(x) dx$ where $f(x)$ is a real-analytic function.
There are many techniques to find that.
Some include contour integration on the complex plane.
But I wonder : Can quaternions be useful for integrals ?
 A: Apologies, this certainly won't be an answer, but I haven't enough reputation to comment. The first I remember hearing of the complex numbers happened in a mathematical statistics course, contour integration is one way of determining a closed form when integrating a probability density function of a normally distributed random variable. It turns out that the theory of complex analysis is surprisingly rich and interesting in its own right, not just as a tool for computing real integrals. As you probably know, complex analysis is intimately related to Fourier series and harmonic analysis. To address your question, I can't say if quaternions are useful for computing real valued integrals, as I don't know of any examples offhand, but they are certainly useful for some integrals due to the existence of the theory of quaternionic analysis. I wanted to mention that other so-called topological groups are useful for integration and have their own variants of the Fourier transform, etc., of which the norm 1 complex numbers (under multiplication) are the prototypical example. If you wanted to explore this further, you might check out the wiki page on Pontryagin duality.
