I have a request on my desk to allow port 35963 through a customers firewall to the pbx. Can someone tell me what this port is for, the guy requesting it cannot tell me.

Could be anything as ports can be changed to be anything. I would get more information.

35963 I don't recognize it but it could be a port chosen for access to DIM, DIM is used for troubleshooting different areas of the phone system functions.
If I were in your shoes I wouldn't open any ports until you know exactly what and why.