Let $\R$ be a skew field. We consider the functional equation \[ F(x)+m(x)G(x^{-1})=0, \quad (\forall x\in \R, x\ne 0), \] where $F, G:\R\to \R$ are additive and $m:\R\to \R$ is multiplicative. We obtain its general solution in the case $\R$ is the quaternions $\mathcal H$ over a subfield of the reals $\mathbb R$. In due course we determine the general form of a quadratic multiplicative $m$ on $\mathcal H$, i.e. solutions of $m(xy)=m(x)m(y), m(x+y)+m(x-y)=2m(x)+2m(y)\, (\forall x,y\in \mathcal H).$ The Euclidean norm $m(x_0+x_1i+x_2j+x_3k)=x_0^2+x_1^2+x_2^2+x_3^2$ is a particular solution. For $G = - F$ $(F(1) =1)$ and $m$ quadratic multiplicative on $\R$ we show that $F$ is not multiplicative and that $F(t^2)$ is not equal to $F(t)^2$ for some $t \in \R.$