Does there exist a quaternion $q$ on the unit sphere such that, given the vanilla complex plane $\mathbb{C}$, $q\mathbb{C}q^{-1} = \bar{\mathbb{C}}$?
Motivation: ordinarily, the plane is rotated by multiplying it by a complex number in the unit circle. Taking the conjugate amounts to flipping the plane along the real axis, which can be viewed as a rotation.