  We study Birkhoff-James orthogonality of bounded linear operators on complex
Banach spaces and obtain a complete characterization of the same. By means of
introducing new definitions, we illustrate that it is possible in the complex
case, to develop a study of orthogonality of bounded (compact) linear
operators, analogous to the real case. Furthermore, earlier operator theoretic
characterizations of Birkhoff-James orthogonality in the real case, can be
obtained as simple corollaries to our present study. In fact, we obtain more
than one equivalent characterizations of Birkhoff-James orthogonality of
compact linear operators in the complex case, in order to distinguish the
complex case from the real case. We also study the left symmetric linear
operators on complex two-dimensional $l_p$ spaces. We prove that $ T $ is a
left symmetric linear operator on $ \ell_p^2{(\mathbb{C})}$ if and only if $ T
$ is the zero operator.
