  Let $L$ be a degree $2$ Galois extension of the field $K$ and $M$ an $n\times
n$ matrix with coefficients in $L$. Let $\langle \ ,\ \rangle : L^n\times
L^n\to L$ be the sesquilinear form associated to the involution $\sigma: L\to
L$ fixing $K$. This sesquilinear form defines the numerical range
$\mathrm{Num}(M)$ of any $n\times n$ matrix over $L$. In this paper we study
the convexity of $\mathrm{Num}(M)$ (under certain assumptions on $K$ and/or
$M$). Many of the results are for ordered fields.
