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
source: The Real and the Complex: A History of Analysis in the 19th Century
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 (1913) by H.F. Burkhardt and. The book is from 1897, the exercises were added in 1913 by S.E. Rasor.