What do I make of a work colleague who whispers my name at random several times a day?

He does seem to have some psychological issues. Based upon what you say here, I would be weary to have anything to do with him. However, like you say, just keep it completely professional and refrain from talking to him about anything other than work. The last thing you want this guy to know is anything about your personal life. If he asks, tell him your personal life is private. Avoid him at all costs, just short of the minimal contact at work.

How do the other people in your office handle this person? Do they totally avoid him, so that you're the only one who puts up with his behaviors? What you could do is ask your supervisor to move your workstation, so that you don't have to be close to him. If he comes over to your work space and attempts to interrupt you, tell him you have a lot of work to do, and you have to get it finished so you won't get into trouble. Then don't respond to him, but appear to work very busily. If he still persists and is annoying, tell him you're going to have to speak to Human Resources because he's keeping you from getting your work done. If he still doesn't take the hint, then do actually report his behaviors to Human Resources. It does sound as if he has a problem.

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Your colleague sounds like he's Schizophrenic.

Go to H.R., tell your boss or tell him to knock it the hell off.