In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called.

As with complex and real analysis, it is possible to study the concepts of analyticity, holomorphy, harmonicity and conformality in the context of quaternions. Unlike the complex numbers and like the reals, the four notions do not coincide.

The projections of a quaternion onto its scalar part or onto its vector part, as well as the modulus and versor functions, are examples that are basic to understanding quaternion structure.

An important example of a function of a quaternion variable is

which rotates the vector part of q by twice the angle represented by the versor u.

The quaternion multiplicative inverse ${\displaystyle f_{2}(q)=q^{-1}}$ is another fundamental function, but as with other number systems, ${\displaystyle f_{2}(0)}$ and related problems are generally excluded due to the nature of dividing by zero.

Affine transformations of quaternions have the form

Linear fractional transformations of quaternions can be represented by elements of the matrix ring ${\displaystyle M_{2}(\mathbb {H} )}$ operating on the projective line over ${\displaystyle \mathbb {H} }$. For instance, the mappings ${\displaystyle q\mapsto uqv,}$ where ${\displaystyle u}$ and ${\displaystyle v}$ are fixed versors serve to produce the motions of elliptic space.