Open additional ports on Azure Web App

No, you cannot. Only ports 80 and 443 are open.

Check this answer on Server Fault: https://serverfault.com/a/751548/394375.

If you need that control, you can use Cloud Services or Virtual Machines.

While Azure Web App cannot expose ports 8080 and 444, you could use Application Gateway to export those ports and route incoming traffic on 8080 and 444 to ports 80 or 443 on a Web App.