This book explores the importance of quaternions and hypercomplex functions in Mathematical Analysis, aiming to provide a systematic and unifying approach to these topics. It covers a wide range of subjects, from the history of quaternions to applications in theoretical physics and cosmology. Suitable for advanced undergraduates, graduate students, and researchers in mathematics, applied mathematics, theoretical physics, and the history of science, it offers a deeper understanding of quaternions and their mathematical foundations, as well as their relevance in understanding the physical universe.