This seems like a poorly thought out question; the IP rights would become the employer's or not based on what was created, when, and how. The domain purchase be

This seems like a poorly thought out question; the IP rights would become the employer's or not based on what was created, when, and how. The domain purchase being public only marginally changes the risk calculus and doesn't change the legal calculus. From a risk calculus perspective, it only matters in the case you actually produce something of value but then becomes a landmine buried deep near the heart of the company, and the risk of that landmine can blow up later things you'd like to do like e.g. accept investment or sell the company. Your competent legal advisers will generally tell you "Do not do things which cloud IP ownership, and in fact, spend material amounts of time and money to do things which uncloud IP ownership such as e.g. getting IP assignments from your employees." The biggest flag I'd raise, as someone who also gets domain fever occasionally, is that buying a domain name is a very tiny amount of the effort involved in a new project.