exp(ρ + θi) = (cosh ρ + sinh ρ)(cos θ + i sin θ)

The angle measure in the above is the θ part. The “linear space” expression of the orientation/rotation is a 2-dimensional quanitity cos θ + i sin θ.

Or if you start with the 2-coordinate version x + iy, where x^2 + y^2 = 1, you can find the angle measure by taking the natural logarithm, imag(log(x + yi)).

Rotations are naturally a multiplicative concept. If you have a rotation a + bi and a rotation c + di, then you can compose them like (a + bi)(c + di) = (ac - bd) + (ad + bc)i (you might recognize this as the the “angle sum” identities from high school trigonometry class).

The idea of the logarithm is that you can convert a multiplicative structure into an additive one. So if you have rotations Q and R, you could normally compose them like QR, but as an alternative you can use a log-space representation (angle measure): QR = exp(log(Q) + log(R)).

A rotation inherently comes with an orientation. The i (“imaginary unit”) is in my opinion properly thought of as a bivector, oriented like a plane. When we express something as a scalar-valued angle measure, we have stripped the orientation out.