Dear Yuri,
Yes - Lubos Motl gave me a difficult time about the fact that E8 (and Garrett Lisi's proposed TOE) does not have complex representations.
In graduate school (in the early 1990's - prior to Super Kamiokande's discovery of Neutrino Oscillations), I learned that the TOE must include complex representations. For that reason, I concentrated heavily on the Special Unitary Groups - I especially like SU(5), SU(7), SU(11) and SU(27). Perhaps I am ignorant and never disected that part of TOE theory, but I thought that Super-Kamiokande's implicated discovery of right-handed neutrinos, and a proper representation for right-handed neutrinos within a theory (Lisi's was nearly correct), negated the necessity for complex representations.
I argued with Lubos until it was obvious that he thought I was crazy, but E8xE8*~SO(32). E8 has 240 real roots plus 8 basis vectors. SO(32) has an order of 496 and has complex representations. SU(11) has an order of 120 and has complex representations.
I reason that either:
1) Lisi's E8 TOE (240 roots) is wrong, and should have been based on SU(11)xSU(11) (120 order times two) with complex representations, or
2) Lisi's E8 TOE (248 order) is wrong, and should have been a Supersymmetric E8xE8*~SO(32) (496 order), where our E8 has exclusively real roots, our E8* has exclusively imaginary roots, and the two are "twisted together" (like a "twistor" algebra) into an SO(32) TOE with complex representations.
Lisi's E8 Gosset lattice implies Octonion Algebra. If we twist a real Octonion together with an imaginary Octonion, then we can generate a real Sedenion (where the progression of Clifford division algebra is: Real, Imaginary, Quaternion, Octonion, Sedenion...).
Peter - Imaginary analysis is part of our mathematical game. Certainly, we must eventually observe real numbers when we perform an experiment, but that does not negate the importance of imaginary numbers.
Besides, e^iPi +1=0 really is cool. Where else can you relate three different kinds of oddball concepts (such as e, pi, and i) into a simple equation?
Have Fun!
Dr. Cosmic Ray