Quaternions aren't new maths - They are so old that Maxwell used them in the first place! ...Heaviside removed them, to simplify things.
http://www.rexresearch.com/maxwell1/20equations.pdf
Thanks for the tip. I did not know about all the work done over the last 150 years to upgrade the Maxwell equations. The Standard model must have absorbed all that theory. Anyone looking to replace all that work is in for a very big job.