Summary: In this paper, a new classification of Sturm-Liouville differential equations with complex coefficients is given. Compared with the corresponding result of B. M. Brown
et al. [ibid. 455, No. 1984, 1235–1257 (1999; Zbl 0944.34018
); Proc. Lond. Math. Soc., III. Ser. 87, No. 2, 419–450 (2003; Zbl 1055.34048
)], this classification reveals the great effects of rotation angle and it is independent of the rotation angles. Moreover, the asymptotic behaviours of functions in the maximal domain are presented and J-self-adjoint extensions associated with the differential equations are characterized.