Some Properties of Quaternion Algebra over the Sets of Real and Complex Numbers

In this article, we analyzed complex quaternions and the matrix representations associated with 2x2 complex quaternions. We provided detailed insights into the fundamental properties of quaternions, highlighting essential features of the corresponding matrix representations. Additionally, we examined real quaternions, emphasizing their specific characteristics, and we explored aspects related to matrix representations of quaternions with real coefficients of size 4x4. The central aim of this article is to conduct a detailed comparison between the two types of matrix representations, both in real and complex contexts. In light of the obtained results, the article seeks to make significant contributions to the understanding and application of quaternions in various mathematical domains.