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According to the Founding King's legend, one with three Guardian Characters will have very special powers and will be given the Humpty Lock to be able to harness those powers. It was this item that gave her the ability to transform, as well as granting the other Guardians this ability later. The Humpty Lock has strong reactions to her emotions.

When Amu Character Changes with a Guardian Character, her hair clip will change and give her specific special abilities.

When Amu Character Changes with Ran, her hair clip becomes a red heart and bestows her athletic and limited flying abilities. A side-effect to this is that Amu becomes honest to the point that she blurts out whatever is on her mind. Amu still has relative control over her body while in this Character Change (unless it a specific task that she has to complete, e.a. swinging around a bar for gym class).
Ran's Character Change is used the most throughout the series.

When Amu Character Changes with Miki, her hair clip becomes a blue spade and gives her increased artistic abilities. However, the Character Change, while short, is powerful and makes Amu spontaneously start drawing to the point that she has no control over her body.
Miki is the one Amu Character Changes with the least.

Su
When Amu undergoes Character Change with Su, her hair clip becomes a green clover and she improves in cooking and housekeeping skills. Like with Miki, Amu does not have much control over her body, to the point that Amu's character literally changes into much more girly version of herself. She is also able to create ingredients out of thin air.
Su's Character Change, while not used very often, is still employed more than Miki's Character Change.

Dia undergoes Character Change with Amu for the first time in Episode 108. When Amu undergoes Character Change with Dia, she becomes a sparkling person with a shiny smile, and calms her down. This change also allows her to release the light inside her heart. Amu's hairclip becomes a yellow diamond when under this Character Change. It doesn't make much of an impression to everyone.
This Character Change only occurs in the anime.

In Chapter 43 of the manga, Il forcefully Character Changes with Amu, granting her a pair of black bat wings and the ability to fly according to Il's command.
This marks the first time she Character Changes with a Guardian Character that's not her own