  We study spectral properties of bounded and unbounded complex Jacobi
matrices. In particular, we formulate conditions assuring that the spectrum of
the studied operators is continuous on some subsets of the complex plane and we
provide uniform asymptotics of their generalised eigenvectors. We illustrate
our results by considering complex perturbations of real Jacobi matrices
belonging to several classes: asymptotically periodic, periodically modulated
and the blend of these two. Moreover, we provide conditions implying existence
of a unique closed extension. The method of the proof is based on the analysis
of a generalisation of shifted Tur\'an determinants to the complex setting.
