Terry Gilliam’s laughter—a high-pitched giggle—is mischievous, playful and serious, much like his films. His latest, “The Man Who Killed Don Quixote,” is notorious for taking nearly 30 years to complete. In 2000, Mr. Gilliam managed just six days of shooting before disasters, including a flash flood and an actor too ill to ride Quixote’s horse, derailed the effort. After struggles to get financing, and legal battles over rights, he made this new version, which opens April 19.

Now 78 years old, Mr. Gilliam became famous decades...