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(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 $\mathbf{Q}$, the group $G(\mathbf{Q})$ is dense in $G(\mathbf{R})$ for the real topology. This is the real approximation theorem for connected linear algebraic groups, see the reference to Sansuc's paper in my comments to 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 $G(\mathbf{Q}_2)$.

(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 $\mathbf{Q}$, the group $G(\mathbf{Q})$ is dense in $G(\mathbf{R})$ for the real topology. This is the real approximation theorem for connected linear algebraic groups, see Sansuc, J.-J. Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. Reine Angew. Math. 327 (1981), 12–80, Cor. 3.5(iii).

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 $G(\mathbf{Q}_2)$.

(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 $\mathbf{Q}$, the group $G(\mathbf{Q})$ is dense in $G(\mathbf{R})$ for the real topology. This is the real approximation theorem for connected linear algebraic groups, see the reference to Sansuc's paper in my comments to 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 $G(\mathbf{Q}_2)$.

(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 $\mathbf{Q}$, the group $G(\mathbf{Q})$ is dense in $G(\mathbf{R})$ for the real topology. This is the real approximation theorem for connected linear algebraic groups, see the reference to Sansuc's paper in my comments to 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 $G(\mathbf{Q}_2)$.

For any connected linear algebraic group $G$ over your field $K=\mathbf{Q}(\sqrt{-3})$, the group of $K$-points $G(K)$ is dense in $G(\mathbf{C})$ for the complex topology. This follows from the real approximation theorem for connected linear algebraic groups, see a reference to Sansuc's paper in my comments to A question on algebraic torus.

(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 $\mathbf{Q}$, the group $G(\mathbf{Q})$ is dense in $G(\mathbf{R})$ for the real topology. This is the real approximation theorem for connected linear algebraic groups, see the reference to Sansuc's paper in my comments to 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 $G(\mathbf{Q}_2)$.