Quaternion Polar Representation with a Complex Modulus and Complex Argument Inspired by the Cayley-Dickson Form

We present a new polar representation of quaternions inspired by the Cayley-Dickson representation. In this new polar representation, a quaternion is represented by a pair of complex numbers as in the Cayley-Dickson form, but here these two complex numbers are a complex ‘modulus’ and a complex ‘argument’. As in the Cayley-Dickson form, the two complex numbers are in the same complex plane (using the same complex root of −1), but the complex phase is multiplied by a different complex root of −1 in the exponential function. We show how to calculate the ‘modulus’ and ‘argument’ from an arbitrary quaternion in Cartesian form.