Personally I would take θ = log(x + yi) / i – where x is a scalar and yi is a bivector, with x² + y² = 1 – to be the definition of angle measure θ of the rotati

Personally I would take θ = log(x + yi) / i – where x is a scalar and yi is a bivector, with x² + y² = 1 – to be the definition of angle measure θ of the rotation x + yi, though usually I like to think of this as θi, a bivector-valued quantity. In that formulation, the structure of rotations is already clear, and the better question might be why the (principal branch of the) natural logarithm of some bivector i is (π/2)i. That starts us on thinking about what the logarithm really means, and asking about conformally mapping a punctured plane to an infinite two-ended cylinder, etc. The reason this formula of Euler’s seems like a mystery is that we teach angle measure to students as the canonical way to represent a rotation which makes it logically primary in their minds, when (in my opinion) it should be a derived quantity. Rotation is a naturally multiplicative kind of operation, a linear transformation of a vector space. If we need to represent a rotation of 2-dimensional vectors by a single parameter, I generally prefer the stereographic projection yi/(1 + x) (“half-angle tangent”) instead of the angle measure (logarithm). Depends a bit on the particular use case though.