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.
In case you consider containerizing and deploying your app:
By default, App Service assumes your custom container is listening on port 80. If your container listens to a different port, set the WEBSITES_PORT app setting in your App Service app. You can set it via the Cloud Shell. In Bash:
az webapp config appsettings set --resource-group <group-name> --name <app-name> --settings WEBSITES_PORT=8000App Service currently allows your container to expose only one port for HTTP requests.