A different kind of imagination: The writer’s muse, daemon, or voice is an alternate personality by virtue of its compelling illusion of independent agency.

A writer’s muse, daemon, voice, character, narrator, etc., are aspects or products of imagination, but they are not aspects or products of what is usually meant by imagination. Here are definitions of what is usually meant by imagination, from the American Psychological Association’s Dictionary of Psychology (2007):

imagination n. the faculty that produces ideas and images in the absence of direct sensory data, often by combining fragments of previous sensory experiences into new syntheses. See also CREATIVE IMAGINATION.

creative imagination the faculty by which new, uncommon ideas are produced, especially when this does not seem explicable by the mere combination of existing ideas. The operations of the creative imagination are sometimes explained by the interaction of dormant or unconscious elements with active, conscious thoughts.

As you can see, ordinary imagination does not involve any additional thinkers with a sense of their own identity. It does not involve a compelling illusion of the presence of another being or self with a mind of its own (“the illusion of independent agency,” in the words of Marjorie Taylor):

Such an illusion is what is meant by multiple personality.