I have just heard of quaternionic analysis. It seems like the goal of this subject is to carry over concepts like homorphicity etc. from complex numbers to quaternions. However, these concepts seem to lose some of the nice properties that we have in complex analysis (eg. derivatives are no longer independent of the path we approach $x_0$ on), so what is the motivation for studying this subject? What are its uses?
Quaternionic analysis as an extension of complex analysis is to my knowledge a field in pure mathematics. And indeed, many nice properties of the complex variable are lost with this structure.
Quaternion algebra however, while a not much interesting field for mathematical research has quite a lot of applications. As @Botond mentioned, Maxwell's equations were originally developed in quaternion form. And as a homage to William Rowan Hamilton, there has been a long tradition of teaching physics with quaternion notation (source). Nowadays, because quaternions provide a singularity-free rotation formalism, they are used in spacecraft attitude control algortihms and in 3D video-games graphics engines. Some applications for pure mathematics may exist of course, such as Lagrange's four-square theorem.