Giuseppe,

As a follower of the Cambridge MRAO and DAMTP groups (e.g. 1st link below, and more generally works by A. N. Lasenby et al), I agree with Alex (whose own book I’ve not yet seen).

And while I'm not sure it will aid your project, if you simply google "quaternionic quantum mechanics" you'll find the subject has a long history.

Some proponents are seduced by the ease of representing the Minkowski metric with 4nions, but most good discussions will guide you back, like Alex does, to the Clifford Algebras (2nd link).

The consensus as regards quaternions, even though they have their proponents, seems to be that they are an interesting variant, but at the end of the day, probably do not themselves offer any real advantages. As the 2nd link says:

 "There is a lot of sexy interplay between the metric geometry using Clifford        algebras and the quaternions.”

One often-quoted paper referring to experiment might be of interest (Brumby & Joshi, 3rd link below).

Also, if you can get hold of a copy, the late Pertti Lounesto's "Clifford Algebras and Spinors", provides a comprehensive background and includes a chapter on 4nions.

Finally, I thank you for your question as it led me to the fourth link, which touches on an interest of my own (spin).

Regards - Paul

 [Notes added later: this last paper - found initially on a CERN site, without author; subsequently located on arXiv; link added), suggests in its opening section, of quaternionic QM, that:

"As a physical theory it should prove to be effective at some high energy level, exhibiting experimental evidence which would distinguish it from the complex theory."

If the suggestion above proves true then it might well be that "high energy" would not be helpful for quantum computing.]

http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/01_anl_erice.pdf

http://math.stackexchange.com/questions/116952/does-quaternion-multiplication-relate-to-minkowski-space

http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/26/070/26070401.pdf

http://cds.cern.ch/record/384318/files/9904067.pdf

https://arxiv.org/pdf/hep-th/9904067v1.pdf

Such basis should be even subalgebra of 3D geometric algebra. Elements of the subalgebra are natural and powerful generalization of complex numbers. See Alexander Soiguine, "Geometric Phase in Geometric Algebra Qubit Formalism" Lambert Academic Press, 2015

Giuseppe,

As a follower of the Cambridge MRAO and DAMTP groups (e.g. 1st link below, and more generally works by A. N. Lasenby et al), I agree with Alex (whose own book I’ve not yet seen).

And while I'm not sure it will aid your project, if you simply google "quaternionic quantum mechanics" you'll find the subject has a long history.

Some proponents are seduced by the ease of representing the Minkowski metric with 4nions, but most good discussions will guide you back, like Alex does, to the Clifford Algebras (2nd link).

The consensus as regards quaternions, even though they have their proponents, seems to be that they are an interesting variant, but at the end of the day, probably do not themselves offer any real advantages. As the 2nd link says:

 "There is a lot of sexy interplay between the metric geometry using Clifford        algebras and the quaternions.”

One often-quoted paper referring to experiment might be of interest (Brumby & Joshi, 3rd link below).

Also, if you can get hold of a copy, the late Pertti Lounesto's "Clifford Algebras and Spinors", provides a comprehensive background and includes a chapter on 4nions.

Finally, I thank you for your question as it led me to the fourth link, which touches on an interest of my own (spin).

Regards - Paul

 [Notes added later: this last paper - found initially on a CERN site, without author; subsequently located on arXiv; link added), suggests in its opening section, of quaternionic QM, that:

"As a physical theory it should prove to be effective at some high energy level, exhibiting experimental evidence which would distinguish it from the complex theory."

If the suggestion above proves true then it might well be that "high energy" would not be helpful for quantum computing.]

http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/01_anl_erice.pdf

http://math.stackexchange.com/questions/116952/does-quaternion-multiplication-relate-to-minkowski-space

http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/26/070/26070401.pdf

http://cds.cern.ch/record/384318/files/9904067.pdf

https://arxiv.org/pdf/hep-th/9904067v1.pdf

Hi Guiseppe, you should really focus on the term "Clifford algebra", there is indeed a Springer journal (AACA) and a couple of quite good books (e.g. Lounesto, Baylis, and some conference AACA proceedings) although nowadays for some people it seems to be fashionable to talk on "Geometric Algebra" which to my opinion is misleading in several aspects... Two important although old (book) publications in this context may be also Hankel's and Rashevsky's books on hypercomplex numbers. In all those notions, quaternions are basic building blocks, however, there are some additional subtleties when relating to geometry (and - please - take care to avoid mixing this up with "Geometric Algebra" and the evangelism of one or the other community around...).

Formerly, the Hamilton quaternion algebra was used in geometry. But then, Heaviside's vectors proved better and were adopted. There is always a way to exploit exotic algebra, but the error should not be made of giving them some ontology. At the end of the day, we always get real observables, and the measurement problem doesn't disappear. There is no real calculation difficulty in quantum mechanics, then little progress can be made in this direction. The standard work is the one of Hestenes with the Dirac / geometric / spacetime / Clifford algebra. 

http://geocalc.clas.asu.edu/pdf/SpaceTimeCalc.pdf

Hi Claude,

Thanks for Pt I of that work - long intended to find it.

If it's not too off-topic: the one thing I've always wondered about Hestenes' work is his Zitterbewegung interpretation of QM.

Any thoughts? 

Regards - Paul 

http://geocalc.clas.asu.edu/pdf-preAdobe8/ZBW_I_QM.pdf

To Paul:

The situation with the Zitterbewegung is perfect demonstration of deep conservatism of scientific community, better to say in words of the "Book of Jungle" monkey philosophy: we all say that, so that's true. Some years ago one researcher from France spent secretly grant money to explicitly demonstrate spinning electron resonance. It was direct demonstration that Zitterbewegung is electron rotation. Only after D. Hestenes insisted, the result was published. And what then? Nothing, absolute silence. Generations of physicists have been talking about  weirdness, mysteries of Quantum Mechanics, inability of human mind to adequately accept quantum world, rejecting any idea that they just work in wrong mathematical formalism. 

Dear Professor, precisely this is what I am checking and prove. We can replace a complex number by a hypercomplex number in a space of dimension three and try to measure the amplitude to know all the cases of superposition of States of the qubit. Thank you for your question, we can work together if you are interested by this subject.

Data Hypercomplex number in three dimensional spaces. hal-01686021

To Alex Soiguine

Dear Alex can you you give a reference to the paper you have mentioned. Thank you in advance.

All the results and analysis of the work of the researchers from France, along with references to their publications, are available from the D. Hestenes text "Zitterbewegung in Quantum Mechanics, Found". It is in open access here, on ResearchGate

The original work of Gouanere et. al is great but the paper  doesn't give that much information. I recommend Rivas instead:

https://arxiv.org/pdf/0809.3635.pdf

And of course Hestenes:

https://arxiv.org/pdf/0802.3227.pdf

In my approach of precanonical quantization in field theory a generalization of quantum theory with Clifford-algebra-valued (hypercomplex) wave functions and operators appears naturally as a result of quantization of Poisson brackets defined on differential forms (within the De Donder-Weyl Hamiltonian formulation of classical fields). Please kindly check my papers on researchgate or type the command "f a kanatchikov" on http://inspirehep.net/

There is a rather down to Earth application of hypercomplex algebra in

viscous fluid mechanics which I like by

Clauser, F. H. 1983, Characteristic modes and fundamental singularities of differential equations, Phys. Fluids, 26, 2787 1983

doi: 10.1063/1.864045

View online: http://dx.doi.org/10.1063/1.864045

View Table of Contents: http://pof.aip.org/resource/1/PFLDAS/v26/i10

Published by the American Institute of Physics.

It shows how to split off the different eigenmodes

Green's functions for viscous flow perturbations.

Gary Webb:

CSPAR, University of Alabama in Huntsville

Better to speak about elements of even subalgebra of the geometric algebra in three dimensions, not hypecomplex numbers. As feasibly follows from my multiple works, particularly available on researchgate, they define much deeper theory than conventional quantum mechanics.

For the attention of everybody:

"Single-photon test of hyper-complex quantum theories using a metamaterial"

https://www.nature.com/articles/ncomms15044

So, maybe an experiment will decide which of the hypercomplex extensions of quantum theory is viable.