Contracts between employers and employees are very common in the US in highly competitive and highly skilled occupations where the employer usually invests a great deal in employees via training, licensing / certs, and other costs.
Non-compete contracts are also common, but are hard to enforce - at least outside of the local area.
I have paid thousands to a former employer (who I left for greener pastures and more money) when I did not stay with the former employer up to an agreed upon date. They paid me a salary, plus paid for my training and testing for several securities and insurance licenses.
I have also paid a fair amount of money to another former employer when I left to start my own business (in the same industry) and took most of my clientele and their assets with me.