The quaternionic Gauss-Lucas Theorem

The classic Gauss-Lucas Theorem for complex polynomials of degree $d\ge 2$ has a natural reformulation over quaternions, obtained via rotation around the real axis. We prove that such a reformulation is true only for $d=2$. We present a new quaternionic version of the Gauss-Lucas Theorem valid for all $d\ge 2$, together with some consequences.