My system:

Lenovo Thinkpad R400
Windows 8.1 64 bit
Intel Core 2 Duo P8400 2.26 GHz
4GB RAM
Intel Integrated Graphics 4500MHD
120GB SSD

I'm particularly interested if anyone has experience of running this game with an Intel integrated graphics chip. My guess is that it's powerful enough but that there may be compatibility issues.

Thanks.

I can't see why not, your CPU is certainly fine. In regards to the onboard GPU, that series is DirectX 10 compatible built-in. The requirements for Anachronox indicate DX9 recommended.

Again there's no guarantees I suppose, but your hardware certainly meets recommendations from what I can see.

Edit: Here's my source to check

http://www.notebookcheck.net/Intel-Graphics-Media-Accelerator-4500MHD-GMA-X4500MHD.9883.0.html

Thanks for the reply. I have DirectX9 runtime installed which allows me to play other DirectX9 games. Will that extend to this?

Yes. Anachronix should be able to recognize and use your pre-existing DX 9 installation.

That is awesome dude. You will not regret it, it's a wonderful game. :)
Glad to hear you got things sorted!