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I don't agree. Complex numbers are the algebraic closure of the reals. Or the quotient of the real polynomial ring by (x^2+1=0). Or whatever other construction. The multiplication rule is the essence of C.

Orthogonality is captured linear algebra over R^2, but R^2 isn't a field or an algebra.

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I think there is still a geometric viewpoint you can bring to the multiplicative structure of C. For example there is the extremely natural homeomorphism between unit C and SO(2). And C minus origin to (R+, SO(2)). It’s completely intuitive for mathematicians to say that 1 and i are separated by 90 degrees.

Indeed, there is a polar coordinates representation of the complex numbers with the nice property that when you multiply you multiply lengths and add angles.

Sniffy nailed it (above) when he mentioned Cayley-Dickson. That's exactly what I was getting at, and had never researched.

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