A complex differential equation is a differential equation whose solutions are functions of a complex variable.
Constructing integrals involves choice of what path to take, which means singularities and branch points of the equation need to be studied. Analytic continuation is used to generate new solutions and this means topological considerations such as monodromy, coverings and connectedness are to be taken into account.
Existence and uniqueness theorems involve the use of majorants and minorants.
Study of rational second order ODEs in the complex plane led to the discovery of new transcendental special functions, which are now known as Painlevé transcendents.
Nevanlinna theory can be used to study complex differential equations. This leads to extensions of Malmquist's theorem.