TITLE: What is the difference between Quaternions and Bicomplex Numbers?
QUESTION [14 upvotes]: So, I know Quaternions are basically 4 dimensional Complex numbers, and the dimensions can double forever to Octonions, Sedinions, etc. I recently heard about bicomplex numbers, which are also sort of 4 dimensional complex numbers. I looked into it a bit, and I found that bicomplex numbers are a commutative version of quaternions. But I wanted to know how exactly that works. How is the multiplication defined? Also can that be extended to 8-Dimensional biquaternions, 16-D bioctonions, etc? And how are those defined exactly? I know with regular hypercomplex numbers, you lose properties as you go up in dimensions, so does the same thing happen with the bi- ones? If so does it just lose the property that it would normally lose one dimension lower? Or do they not lose properties at all? Any answers would be greatly appreciated.

REPLY [2 votes]: Bicomplex numbers, or tessarines is a commutative algebra as other people said.
It has zero divisors. And I would say, being commutative is a lot more important than disadvantages from zero divisors!
Moreover, zero divisors are not a disadvantage at all, because it allows bicomplex numbers to generalize not only the concept of complex numbers, but also hyperbolic (split-complex) numbers. They are included in bicomplex numbers and have zero divisors.
There is nothing essentially bad regarding zero divisors: this means that we have some elements by which we cannot divide, but we anyway had one such real number: zero.
Unlike what the other answer said, of course there is norm (modulus) in bicomplex numbers.
They can be represented as real matrices of certain structure with usual matrix operations, so one can experiment with them in any computer algebra system that allows to apply analytic functions to matrices.


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