Algebraically, and also in group theory, it's exactly that. A complex plane is re-interpreted as a half-way mirror operation on an orthogonal axis or quadrature

Algebraically, and also in group theory, it's exactly that. A complex plane is re-interpreted as a half-way mirror operation on an orthogonal axis or quadrature plane (eg the imaginary axis in the case of the 2D Argand plane). Which is what you you get when you multiply by i in complex numbers or i/j/k in quaternions - a 90 degree rotation. The correct way to do arbitrary rotations in this case is to use an exponential process, and then you get Euler's equation, the quaternion symmetry operation or, in general, the exponential map of an infinitesimal transformation in a Lie group. In higher dimensions you get other type of (sometimes weird) operations, related to the Cartan–Dieudonné theorem.