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We present a new polar representation of quaternions inspired by theCayley-Dickson representation. In this new polar representation, a quaternionis represented by a pair of complex numbers as in the Cayley-Dickson form, buthere these two complex numbers are a complex 'modulus' and a complex'argument'. As in the Cayley-Dickson form, the two complex numbers are in thesame complex plane (using the same complex root of -1), but the complex phaseis multiplied by a different complex root of -1 in the exponential function. Weshow how to calculate the amplitude and phase from an arbitrary quaternion inCartesian form.

Quaternion polar representation with a complex modulus and complex argument inspired by the Cayley-Dickson form