> Is there any way to hide my client IP when I connect to the database?

There's no easy way to hide the client side IP if you connect to the database 
through SQL*Net (called Oracle Net now).

However, if you have permission, you can telnet or ssh to the database server 
and connect to the database bypassing SQL*Net. Then 
sys_context('USERENV','IP_ADDRESS') will be null. Alternatively, you can set 
up NAT (network address translation) on your side. Then when you connect to 
the DB, IP_ADDRESS of USERENV would show the NAT IP.

If you use Windows, turning off Messenger service prevents others from seeing
your logon name by nbtstat -A <your IP> | find "<03>".

Yong Huang