  Let $p$ be a multilinear polynomial in several non-commuting variables with
coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$,
the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either
zero, or the set of scalar matrices, or the set $sl_n(K)$ of matrices of trace
$0$, or all of $M_n(K)$. This conjecture was proved for $n=2$ when $K$ is
closed under quadratic extensions. In this paper the conjecture is verified for
$K=\mathbb{R}$ and $n=2$, also for semi-homogeneous polynomials $p$, with a
partial solution for an arbitrary field $K$.
