Publishing outside of my PhD Research
This is something that you need to discuss specifically with your advisors. They may or may not be interested in the publication—but they may also have an interest in ensuring that your goal of publishing doesn't interfere with your paid work. (For instance, spending a lot of time editing your "hobby" manuscript when you should be working on your talk for an upcoming conference for your "paid" project.)

You should also check with your university's intellectual property office about the guidelines for such work. So long as you're doing things on your own without university assistance or resources, there might not be a problem. But if you need to use equipment owned by them to produce your idea, things might change.