Is there a additive non-linear non-complex map? 
Let $V$ and $W$ be vector spaces over a field $\mathbb{F}\neq \mathbb{C}$. Give an example of a non-linear map $T:V\to W$ such that \begin{equation}T(x+y) = T(x)+T(y), \forall x,y\in V.\end{equation}

I asked myself this question when I was resolving an excercise list os Linear Algebra. This example is pretty easy when $\mathbb{F}=\mathbb{C}$. We take $V=W=\mathbb{C}$ and $T:z\mapsto \bar{z}$, and we have that $T(\lambda z) = \bar{\lambda}T(z)$. However, I couldn't find any examples for non-complex vector spaces.

A discontinuous additive function $a\colon\Bbb R\to\Bbb R$ is non-linear map of a linear space $\Bbb R$ over a field $\Bbb R$ to itself. Observe that any additive map is linear, if the field of scalars is $\Bbb Q$.
