Given $\omega_1,\omega_2\in\mathbb{C}\setminus\{0\}, \frac{\omega_1}{\omega_2}\notin\mathbb{R}$, define $L(\omega_1,\omega_2)=\{n\omega_1+m\omega_2:n,m\in\mathbb{C}\}$. The torus associated to $L(\omega_1,\omega_2)$ is $T_{\omega_1,\omega_2}=\mathbb{C}/L(\omega_1,\omega_2)$. Now $T_{\omega_1,\omega_2}$ has the natural complex structure: for $x\in T_{\omega_1,\omega_2}$, fix $z\in \mathbb{C}$ such that $\pi(z)=x$, where $\pi:\mathbb{C}\rightarrow T_{\omega_1,\omega_2}$ is the natural quotient map; choose $D$ to be an open sufficiently small disc centered at $z$, and then $(\pi(D), (\pi\restriction_D)^{-1})$ is a chart at $x$; it may be checked that charts of this form are compatible so that we get a complex structure on $T_{\omega_1,\omega_2}$.
I wonder whether it is true that any complex structure on an abstract real torus arises in this way? By "an abstract real torus" I mean a real 2-dimensional surface of genus one. I guess it is obvious that any such torus is homeomorphic to the topological space $\mathbb{C}/L(\omega_1,\omega_2)$ (with the quotient topology) for some $\omega_1,\omega_2\in\mathbb{C}\setminus\{0\}, \frac{\omega_1}{\omega_2}\notin\mathbb{R}$, but is it obvious that one cannot put a complex structure on $T_{\omega_1,\omega_2}$ apart from the natural one described above?