Could I say that a complex number can be visualized as a point in 2-D space with one coordinate beeing the real part and the other the imaginery part?
Absolutely! But remember that this 2D space is not a subspace of our known 3D space; it takes 1 dimension from our 3D space (the real part) and one from another, imaginary space.
And the quaternion would then be “visualized” 😉 as a point in 4D-space on a 4D-hypersphere?
NO! (Technically: I could say that, but I wouldn’t(, because it is very likely to be misunderstood and nobody knows what a 4D hypersphere looks like anyway) and you can’t say that…)
It can be seen as a point in a 4 dimensional space, but that is not our 3D plus one more dimension (something like spacetime is our 3D world plus one…)!
It is an imaginary 3D space plus one real dimension!
To say something like: “A complex number is a 2D number” (and I know I’ve said that myself…)is actually short for something like: “A complex number is an element of C (the collection of complex numbers) and C is homeomorph to R (the collection of real numbers) * R = R² and can therefor be represented as a 2 dimensional vector.”
Homeomorph here means that there exists a one-to-one mapping between the two.
In (mathematical) topology we say that for instance a donut is homeomorph to a coffeecup (you can think of topology as ‘geometry without a distance’. If two points are ‘connected’ in one, then they are ‘connected’ in the other), although they are obviously not the same (try pouring coffee in a donut…).
My advice: Do not try to visualize a quaternion and save yourself a truckload of problems!
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