THIS BOOK: WHY DOES IT MATTER? This book leads essentially with quaternions and functions of quaternions. Roughly speaking, hypercomplex may be considered as a living subject in the field of Mathematical Analysis, in the sense that a systematical and unifying approach for dealing with them is still missing.In this book we will try to fill that gap. Growing out of a course designed to teach “Complex Analysis” to honors- level undergraduates of Mathematics and Applied Mathematics at the UNESP (São Paulo State University), this Book is suitable for use as a textbook in a course or self-study by advanced undergraduates or graduate students who possess a basic familiarity with Complex Analysis. This book covers a broad variety of topics ranging from a comprehensive overview on the history of quaternions, and a consistente foundation for series, derivation, and integration of hypercomplex., with an exposure of a Cauchy – like Theorem. From a comprehensive account of quaternionic series, some selected applications on theoretical physics, such as topics on classical field theory and cosmology, focusing on some original contributions by the authors (e.g., the coupling between electric and gravitational fields and a brief presentation on “ Cosmological Cycles of Time”: an investigation about a quaternionic version of the “Cosmological Cycles of Time in modern Cosmology.) are presented. It is our purpose that the reader shall come away with a better knowledge on quaternions, their mathematical foundation and applications, and how these geometrical objects may play na essential role on the understanding of our physical universe. This book may also be suitable for graduate students and researchers working in mathematics, applied Mathematics, theoretical physics, and in the history of science.