  In this paper we study biconservative immersions into the semi-Riemannian
space form $R^4_2(c)$ of dimension 4, index 2 and constant curvature, where
$c\in\{0,-1,1\}$. First, we obtain a characterization of quasi-minimal proper
biconservative immersions into $R^4_2(c)$. Then we obtain the complete
classification of quasi-minimal biconservative surfaces in $R^4_2(0)=\mathbb
E^4_2$. We also obtain a new class of biharmonic quasi-minimal isometric
immersion into $\mathbb E^4_2$.
