Complex numbers and (Pauli/Dirac) matrices not required if you use Geometric Algebra. I highly recommend the book by Doran and Lasenby [0], or you can get the details from their papers, notably [1].

[0] Geometric Algebra for Physicists, CUP, 2003

[1] https://arxiv.org/abs/quant-ph/0509178

https://deferentialgeometry.org/papers/Doran,%20Lasenby%20-%...

page 28, equation 2.36. Thanks a lot I'll take a dive into this

Note: my inquiry was motivated by this:

https://blog.revolutionanalytics.com/2014/01/the-fourier-tra...