I just thought of a frequency-multiplication instead of a -addition (per partial!!) at first and it just didn't sound like a multiplication...
It is possible to multiply frequencies per partial if you manage to isolate them sufficiently. Instead of direct complex multiplication, you need to do complex division and compute instantaneous amplitudes and frequencies (analysis), multiply the frequencies, and recreate the signal (resynthesis). See attached patch for a demo. Real world signal tend to have so many partials however, that the method is impractical.
..I wonder if there are also "hyper-analytic" signals utilising quaternions..
It seems that quaternionic and other hypercomplex signals are mostly used in image processing and other 2D / 3D processing. How could one create more than two orthogonal phases out of a 1 dimensional (audio) signal? And, if possible at all, what information could one derive from the extra phases? Intriguing matter...
By the way, analytic signals are normally not considered very useful in audio processing either, except for single sideband modulation in radio transmission. Yet I use analytic signals in my Pd setups for a lot of purposes. For example, to find amplitudes for an ultra fast compressor / expander ([qompander~], shared on this forum).