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>The complex numbers are not a two-dimensional Euclidean space directly, but are a space of transformations (scaling & rotation) on two-dimensional Euclidean vectors, where 1 represents the identity transformation, and i represents a quarter turn anticlockwise.

You might be right, I don't know. Could you explain a bit more how you mean?

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Complex numbers have multiplication defined on them. If you multiply two complex numbers, you get another complex number. (They compose via multiplication in exactly the same way as scaling & rotation operators on 2-dimensional vectors.)

If you just have 2-dimensional vectors, there’s no obviously well-defined way to multiply two vectors and get out another vector.

In other words, both 2-dimensional vectors and complex numbers are made up of 2 coordinates, but they don’t have the same mathematical structure.

I suspect the important point is that rotations cycle around (keep rotating and you come back to where you started)

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