Yes, you can extend your stay in the UK after the completion of your degree. In fact, after successfully completing an undergraduate or Master's degree, international students may request to stay in the UK for up to two years. Again, graduates with a PhD may seek a three-year extension. However, ensure that your visa extension is done with proper documentation. Else, often, you may get penalised. That's why it's better to let your professors keep you in the loop as well.