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Complex numbers in 2D, quaternions in 4D, why nothing in 3D? 

I'm just trying to understand why quaternions are necessary. If I understand right, first off please check me on this, Hamilton kept trying to find ways to multiply triplets and found something satisfactory only when he escalated to four dimensions. 

So game coders and spacecraft engineers extend a vector into four dimensions, then rotate it by a conjugation that takes it out of 3-D space with the first multiplication and pulls it back with the second. (And please check me on that too). 

Is there an intuitive way to understand why that's necessary? Or is it one of those things that can only be explained formally?