Any ethical problems with dating a former student?
I don't see any inherent ethical problems here: the key word is former student, so there is no present power dynamic, conflict of interest or anything like that.
However, you should be prepared for those in your department to find out that you are dating a former student, which could create a certain amount of push back. For instance:
The one year gap between the student being in your course and your romantic relationship will not be apparent to all who know about the relationship. Even if you tell them, they may not believe it. Even if neither of you expressed an interest in dating until recently, people may still suspect that you had romantic feelings for one of your students while they were your student.
There is presumably some age differential. If it is small enough, people will probably ignore it. If it is large enough, a lot of people will try to figure out "Why is s/he dating him/her?" This will motivate them to find out that your romantic partner is your former student.
If you date more than one former student, people may view you as using your classroom as a hunting ground for future romance. If this sentiment is held widely, it could make your future students uncomfortable.
I hasten to add that none of these point to any clear ethical lapse or mistake on your part. In particular, the idea that it is disturbing for a much older (adult) to date a much younger (adult) may be more of a societal taboo/hangup than anything that can be rationally justified. It is also true that most romantic relationships come with a certain amount of push back, and some kinds of romantic relationships still get a lot of push back for things that many/most of us feel strongly are absolutely unproblematic: e.g. inter-racial relationships, same sex relationships. (In fact, I am very sorry to say that if your relationship is not inter-racial or same sex, any push back you get will probably be mild compared to that, and if it is, that will increase the push back considerably.) I recommend weighing the possible costs of such a relationship against the possible benefits.
The professor-student relationship does not end when the course ends, and indeed might not end a year later. Professors are asked to write letters of recommendation for former students. Sometimes they publish papers with their former students as coauthors, based on work they previously did together. Further, if "student" means that this student completed a thesis under OP's direction (and not just took a class from OP), then this relationship is a lifelong one.
So long as the professor-student relationship continues to exist, a potential power imbalance exists, and an ethical problem arises. Whether your colleagues consider this a serious ethical problem is addressed by Pete Clark's answer.
No. This is not unethical. Consider the alternative.
If I am single, I reserve the right to pick off a member of society to be my mate. The very notion that I must temporarily restrain myself from choosing such a relationship with the people whom I actively have direct authority over... is an idea that makes some sense. The idea that anyone is permanently blacklisted from being a potential candidate, just because I have ever encountered that person in a class which I taught, is way too unfairly exclusionary. Such a requirement would be unethical.
I remember a college class which was required for all students in the college. Would it be a sensible expectation that a college professor should be required to seek a mate from among the less educated, or from a remote town?
When I signed up to be a college professor, I never agreed to limit my long-term life options in such a way. Once I am done with the short-term scenario where I have influence over a person, such an expectation would impose notable cost/harm without really providing significant benefit.