  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 amplitude and phase from an arbitrary quaternion in
Cartesian form.
