Per the title, what are some of the oldest complex analysis books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am aware of the classic books of Goursat and Titchmarsh.
It seems the first textbook on complex analysis was J.C. Bouquet and C.A. Briot, 1859, Théorie des fonctions doublement periodiques et, en particulier, des fonctions elliptiques
This 1859 text contains "examples", but no unsolved exercises. An early 20th century textbook with unsolved exercises is Theory of functions of a complex variable by H.F. Burkhardt. The book is from 1897, the exercises were added in 1913 by S.E. Rasor.
One of the first English complex analysis books with (unsolved) exercises was Whittaker and Watson, Course of Modern Analysis. First edition was published in 1902 (by E. T. Whittaker only). The exercises (more than 1000 of them) are mostly taken from the famous Cambridge Tripos exams. Some of them are very hard.
This book is famous for other reasons too: Old books still used
Remark. It seems to be an English-American custom to include exercises in textbooks. In other languages (Russian, French) they don't include them, and publish separate books of exercises.