Alternative to imaginary numbers? 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? 
 A: 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:


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*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:


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*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:


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*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.
A: I don't know what "other means" the fellow has in mind, but there are a couple of ways to do what complex numbers do without ever introducing imaginary units: 


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*If you know about rings, ideals, and quotient rings, then you can work in ${\bf R}[x]/(x^2+1)$ which has an element, $x+(x^2+1)$, which does whatever you want your imaginary unit to do. 

*If you know about matrices, the set of all matrices of the form $$\pmatrix{a&b\cr -b&a\cr}$$ with $a,b$ real does everything you need, with $\pmatrix{0&1\cr-1&0\cr}$ playing the role of the imaginary unit. 
A: The following is his response. 

Hi,
I was alluding to Clifford Algebra (some people call it geometric algebra).
  See a paper by Chris Doran, Stephen Gull and Anthony Lasenby with a title something like "Imaginary Numbers are not Real ..." 
The complex numbers are a sub-algebra of the simplest of all Clifford Algebras, Cl_2.
  Moreover the "vector" nature of complex numbers is captured by the complementary sub-algebra.
SDG

A: Complex numbers are often great explainers and illuminators. Here is a canonical example. We have
$${1\over 1 + x^2} = \sum_{k=0}^\infty (-1)^n x^{2n}.$$
A bright calc student will be prompted to ask, "What is the deal here?  Why does the series suddenly stop converging at $\pm 1$?  The function on the left-hand side is differentiable to any order on the entire line."   
The complex plane reveals the answer.  The function $f(z) = 1/(1 + z^2)$ has poles at $\pm i$.  So, the distance from the center of the Taylor series to the place where it first has an analytical nasty (a pole here) is 1.  All of a sudden, this mysterious "stoppage of convergence out of the blue" becomes an entirely natural phenomenon. 
I fail to see merit in this guy's idea that complex numbers are somehow unnatural.  
A: Maybe he meant the following: A complex number $z$ is in the first place an element of the field ${\mathbb C}$ of complex numbers, and not an $a+bi$. There are indeed structure elements which remain hidden when thinking in terms of real and imaginary parts only, e.g., the multiplicative structure of the set of roots of unity.
