As with any paranormal investigation, we must begin looking for answers with the most likely or logical explanations. What your fiancé experienced was probably a phenomenon called the hypnagogic state -- a state of "threshold consciousness" between wakefulness and sleep. In this state, people can have "waking dreams"; that is, they are dreaming in a half-awake state. This can result in confusing hallucinations that they think are real.
What makes me think that this is what your fiancé experienced is that he did not react immediately to you being there when he knew you were at your home. His vision of your "double" was really a waking dream.
Paranormal explanations are less likely:
- It was your double, or doppelganger -- whatever that might mean
- It was a psychic projection: In your desire to care for your fiancé, you psychically projected an image of yourself to his bedside; some researchers might say you were astrally traveling
- It was a ghost or some other kind of entity that he only thought was you.