TITLE: Alternative to imaginary numbers?
QUESTION [9 upvotes]: In this video, starting at 3:45 the professor says

There are some superb papers written that discount the idea that we should ever use j (imaginary unit) on the grounds that it conceals some structure that we can explain by other means. 

What is the "other means" that he is referring to?

REPLY [5 votes]: Seamus Garvey is making a lot of sense here and alluding to something pretty deep.  Geometric algebra is a unifying concept that can seem like magic for someone who has never seen them before.  It scoops up complex numbers, quarternions, exterior algebra, spinors, and host of other tools that previously seemed unrelated.  
To see their use in physics check out:

David Hestenes has been a long time proponent and uses geometric algebra to push an undergrad course on mechanics way past what can usually be done for undergrads:
http://www.amazon.com/gp/product/0792355148/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0521480221&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=06N9652VYPQXDFB81SYD
Baylis takes the standard course on electromagnetism at the undergrad level and rephrases it in terms of geometric algebra (arguably a much more natural approach): http://www.amazon.com/gp/product/0817640258/ref=pd_lpo_k2_dp_sr_2?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0470941634&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=1J8N3BGYNA8465XY6XV5

and to see a mathematical approach that leads you down the spinor path check out:

Lounesto's http://www.amazon.com/Clifford-Algebras-Spinors-Mathematical-Society/dp/0521005515/ref=sr_1_1?s=books&ie=UTF8&qid=1309116555&sr=1-1

or something a bit older and explicit is:

Altmann's http://www.amazon.com/Rotations-Quaternions-Double-Groups-Altmann/dp/0486445186/ref=pd_sim_b_8

I can't stress enough how the Clifford Algebra concept brings it all together.  One works with all these separate tools and you get a feeling that it's all related but it's not often that they're presented as such.  Now don't get me wrong -- Clifford algebras are not a magic bullet and complex analysis will always be in your tool box.  That's a fact. But to see how it all links up go the CA route.  
You won't be sorry.