Square roots of complexified quaternions

TL;DR

This paper investigates the square roots of complexified quaternions, exploring their forms and existence using Clifford algebra isomorphisms, revealing diverse root structures including discrete, continuous, or nonexistent roots.

Contribution

It introduces a method to find square roots of various complexified quaternions using Clifford algebra isomorphisms, highlighting their diverse root structures.

Findings

Complex quaternionic roots can be discrete, continuous, or absent.

Isomorphisms with Clifford algebras facilitate root computation.

Examples demonstrate the variety of root structures.

Abstract

Square roots of complexified (complex) quaternions, namely, the Hamilton quaternion, coquaternion, nectorine, and conectorine are investigated. The isomorphisms between the complex quaternions and 3-dimensional multivectors of Clifford algebras is employed for this purpose. Root examples for all named quaternions are presented from which follows that the complex quaternionic roots may assume discrete or continuous form, or there may be no roots at all.