I am thinking about the ‘complex’ transport equation: $$\partial_t + a\cdot \nabla = 0$$ where $a \in \mathbb{C}^n$ and its elements are either real or pure imaginary. The simplest example for this operator is $n=1$ and $a\in \mathbb{R}$, which is then simply the 1D transport equation: $$\partial_t + a \partial_x = 0.$$ The simplest, more general operator in $n$ dimensions would have $a = (x_1, ..., x_n)$ where $x_i \in \{\pm1,\pm i\}$. For example, in 3D we might have: $$\partial_t + (1, -i, -1) \cdot\nabla = 0.$$
I have not been able to find general treatments of such operators in my search, only vague references that are not so helpful, for example here. What are these operators called, and where can I find the general theory of them?