Causal and self-dual morphisms in four complex dimensions

We define a class of maps between holomorphically embedded null curves which generalize conformal transformations, and can be defined in any complex dimension. In four dimensions, we can also define a similar map between self-dual surfaces, which generalize flat $\alpha$-planes. These maps are respectively called causal and self-dual morphisms. It is shown that there exist an infinite class of non-trivial examples for both types of maps in four dimensions.