If they're asking you to do this on your work computer, I'd do it and be done with it. Bottom line is, it's their hardware. If they want to monitor what's happening with their computers, there are many ways to do it, and it's in the company's interest.

I find it a little odd, though. As Nick says, there's a good probability that your IP addy today might not be your IP next week. Are your IPs allocated outside your company? If not, your IT boys should be able to determine anyone's IP at any given time.

(Actually, the more I think about it, the stranger it seems. Do you have any kind of IT dept there at all? Why don't they configure all their computers to report the machine's IP every so often? There's more than one way to do it. Automated, silent, no fuss, no muss.)

But...if they're telling you to do this with your personal, home computer, the only reasonable response is "pound salt."* Besides being none of their damn business, it's of little practical use to them. Let's say they were planning to allow you to get into the company intranet at home, with your access based on your IP. Again, there's the problem of dynamic IPs to contend with. Granting access via a user id and password (as does the rest of the world) would make more sense.

*Unless you connect from home to a company VPN, in which case you're using their resources again.

Damn, I'm rambling again. Sorry.