4-dimensional generalization of the continued fractions

Let $L_i(X), i=1,2, L_1 \ne \bar{L}_2$, be two complex linear forms in $\mathbb{R}^4$, and $K_i(X)=L_i(X)\bar{L}_i(X)$ are positive quadratic forms. The root sets $\mathcal{L}_i$ of forms $K_i$ are two-dimensional planes in $\mathbb{R}^4$. Assume that $\mathcal{L}_1 \cap \mathcal{L}_2=0$ and that there are no integer points except $0$ which lie at $\mathcal{L}_i$. We propose an algorithm of computation of integer points that give the best approximations to the sets of roots $\mathcal{L}_i$. If coefficients of forms $L_i$ lie in totally complex quaternary conjugated number fields $k_i$, our algorithm often finds unities of $k_i$. The algorithm was tested on the set of quaternary number fields specified by equations with small coefficients. The algorithm was successful more often than the best of known algorithms in totally real quaternary case — the Güting algorithm.