On Interactions of Quaternionic and Complex Structures of Linear Spaces

On various occasions, when working with quaternionic linear spaces, there is a need to restrict them to their complex linear structure, then it becomes essential to understand whether the pre-existing internal product or norm in the quaternionic space will continue to be compatible with the complex structure of the new space obtained. There are other situations in which these types of questions arise, for example, if a linear space is originally complex but it turns out that it also admits the quaternionic structure. The objective of this work is to present the different options to change the linearities of some linear spaces and to analyze what happens with the pre-existing algebraic objects: to understand if they still work or if they induce some others that will be compatible with the new linear structure.