I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs.

Mostly, I'm just looking for definitions and basic existence and uniqueness results. Online results, from what I've found, just "leaped" to advanced or specialized topics.

Old classical textbooks cover both real and complex case (with emphasis on the complex case). One of the best is

E. L. Ince, Ordinary differential equations (multiple editions).

Others are

E. A. Coddington and N. Levinson, Theory of Ordinary differential equations (multiple editions),

E. Hille, Lectures on ordinary differential equations, Adison Wesley, 1968.

(This is a different book from the one recommended by Igor Khavkine, and I like it more).

A classic reference is

Hille, Einar, Ordinary differential equations in the complex domain, Pure and Applied Mathematics. New York etc.: John Wiley & Sons, a Wiley-Interscience Publication. XI, 484 p. (1976). ZBL0343.34007.

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