>>>> That's also why, even if complex cannot be extended in 3D, they can in
>>>> 4D. (look at quaternion...)
>>>
>>> No. Quaternions do not form a field. Neither do the hypercomplex
>>> numbers, nor any of the other 4D generalisations.
>>>
>>
>> That's irreverent to Le_Forgeron's point. The application of the hairy
>> ball theorem here does not depend on the numbers forming a field.
>
> I should clarify, I took the fact that the extension wouldn't be a field
> to be implicit in Le_Forgeron's point, since the Hairy ball theorem's
> use doesn't depend on the numbers forming a field. Not saying your
> reading was technically incorrect.

By "extend the complex numbers", I implicitly meant to extend it to make 
a larger /field/. Apparently that is impossible. There are of course 
several ways you can extend them if you don't mind the result failing to 
be a field...

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