I think it's up to the co-worker to handle. If he starts on you, you can just let him know that you don't appreciate personal nosy questions. You'll have to be very firm in not answering them. You can usually train someone to stop doing something you hate, or at least find a way to minimize your contact with them while still refusing to answer their nosy questions. If he persists and you absolutely can't stop him, sure, complain to someone in authority over him. It's best if you have to do that to show what you have done on your own to solve the problem.
Not to single out any particular poster, but I notice that so many answers given in this part of the forum contain the advice "go to HR." More people are employed by small companies than big giant conglomerations that have HR departments. Sometimes I feel that advice is unhelpful. The spirit of it, going to someone higher up who is equipped to handle it, is correct. But the reality is, in many small businesses, the offender may be the owner, or a family member of the owner. Even if not, office politics come into play and it's tricky to negotiate some of these situations. Especially if you must keep the job, and continue to get along with not only the offender, but other workmates who may not see the issue from your point of view.
The bottom line is, while no one should have to put up with being harassed, I feel that sometimes the "go to HR" answer is a little naive. As though at every company there is some special department set aside to magically resolve issues for grown adults in the workplace. The truth is, very few companies have departments like that, and it's not very practical for employees to always be calling government labor boards or lawyers or writing threatening letters, etc. There are some things that are just unacceptable and have to be fought. But in my experience, in a real world workplace, negotiating these issues can be very delicate and it's usually best for the employee to find a way to manage it on their own, without looking like a whiner or a problem child to the other adults they are working with.
Sorry, I just had to get that out. There are some issues that DO require the law to step in, or some outside entity. But I think it's almost always better for the worker to find a way to handle it themselves if possible. It's just another part of getting along in the adult world. I feel like the legitimate issue of harassment has been weakened to sometimes mean, "I don't like it, somebody make it stop." 90% of the time, if you don't like the way someone is treating you, it's in YOUR power,and your best interests, to be the one to make them stop.