(EDITED taking in account the comment of Yves.) The group denoted in the question by $SU(2,1)(K)$ is the group of $\mathbf{Q}$-points $G(\mathbf{Q})$ for a suitable $\mathbf{Q}$-group $G$, and the group denoted by $SU(2,1)(\mathbf{C})$ is $G(\mathbf{R})$. For any connected linear algebraic group $G$ over your field $K=\mathbf{Q}(\sqrt{-3})$, \mathbf{Q}$, the group of $K$-points $G(K)$ G(\mathbf{Q})$ is dense in $G(\mathbf{C})$ G(\mathbf{R})$ for the complex real topology. This follows from is the real approximation theorem for connected linear algebraic groups, see a the reference to Sansuc's paper in my comments to http://mathoverflow.net/questions/87665/a-question-on-algebraic-torus.
Note also that if $G$ is a simply connected semisimple group over $\mathbf{Q}$ (as the group in the question), then $G(\mathbf{Q})$ is dense in $G(\mathbf{Q}_p)$ for any prime $p$. This follows from the weak approximation theorem for simply connected groups (due to Kneser, Harder, Platonov). This is not true for semisimple groups that are not simply connected, see example 5.8 in Sansuc's paper, where a certain semisimple $\mathbf{Q}$-group $G$ is constructed for which $G(\mathbf{Q})$ is not dense in $G(\mathbf{Q}_2)$.
