Hörmander's theorem says that if $L = \sum _{i=1} ^r X_i ^2+ X_0 + f$ on some open subset $U \subseteq \Bbb R$ has the property that the Lie algebra generated by $\{X_0, \dots, X_r\}$ at every point has real dimension $n$, then $L$ is hypoelliptic. Here, all vector fields are assumed to be real.
Is there any version for the case when the vector fields are complex? A quick look at Marco Bramanti's "An Invitation to Hypoelliptic Operators and Hörmander Vector Fields" doesn't seem to address this.
You will find a study of self-adjoint operators of type $$ \sum_{j=1}^r(X_j^*-iY_j^*)(X_j+iY_j), $$ where $X_j, Y_j$ are real-valued vector fields for instance in the Helffer-Nier Lecture Note (Lecture Notes in Mathematics, 1862, Springer-Verlag, 2005). The geometry of complex vector fields is much more complicated than in the real case. Just to give a simple example, the 2D vector fields $$ \partial_{x_1}+ix_1^{2k}\partial_{x_2} $$ are hypoelliptic as well as their adjoints, although $$ \partial_{x_1}+ix_1^{2k+1}\partial_{x_2} $$ is micro-hypoelliptic at $\xi_2=1, \xi_1=0, x_1=0$ and not micro-hypoelliptic at $\xi_2=-1$, $\xi_1=0$, $x_1=0$.
On the other hand, L. H\"ormander is studying non-self-adjoint pseudo-differential operators in Chapter 27 of the fourth volume of his treatise ALPDO, a nice reading.
As Bazin notes, the situation is more complicated for complex vector fields.
For example, Kohn (Annals of Mathematics, 162 (2005), 943–986) gave an example of an $L^2$ sum of squares of complex vector fields satisfying Hormander's condition, which is hypoelliptic but fails to be subelliptic. He has examples with a loss of as many derivatives as you like.
Christ, based on Kohn's example, showed that the $L^2$ sum of squares of complex vector fields satisfying Hormander's condition can fail to be hypoelliptic.
