Pulse shallot in a food processor until finely chopped, about 12 pulses. Add salmon; pulse until mixture is finely chopped, 8 to 10 pulses. Transfer half of the mixture (¾ cup) to a large bowl, and set aside. Add mayonnaise, sour cream, cream cheese, lemon zest, and lemon juice to remaining salmon in food processor; process until mixture is smooth and creamy, about 20 seconds, stopping occasionally to scrape down sides of bowl.
Transfer creamy salmon mixture to finely chopped salmon mixture in large bowl. Add chives, dill, and capers; gently fold ingredients together until incorporated. Season to taste with salt and pepper. Transfer dip to a serving bowl. Garnish with additional chives and dill. Serve.
