I would love to be able to login to Elite : Dangerous as another Commander.
Is there a menu option I'm missing or is it as simple as registering another account on the website?
You only get one active commander per purchased account, so you'll need to buy another copy of the game (presumably on a new store account) and then log into the launcher with the new account.
As mentioned already, you need a second account, as multiple commanders are not supported on a single account yet, but a recent post on the Elite Dangerous forums confirmed that they are still planned.
If you do go for the "additional account" option and are going to sign in to both accounts on a single machine, I have found that it gets a bit annoying as every time you switch account on the launcher, you have to receive the verification code and type that in.
There is a way round this: locate the folder that your launcher is installed in - right click the icon and select "Open file location", the folder is likely called "EDLaunch". Make a copy of this folder for the additional account, then run the "EDLaunch.exe" file within and enter the second account login details.
This way, you can run each copy of the launcher individually without always having to change login details and enter the verification codes. You can make a shortcut to the second launcher EDLaunch.exe file and name it whatever you like to help identify it.
One thing to note: When you update the original install of the launcher with a new version of the launcher, you'll need to delete the copy you made and then copy the new install as before. If you keep the folder name the same for the new copy, you won't need to re-enter any login details or change your shortcut.
Edit: Reference to confirmation from the developers that Multiple Commanders are still on the roadmap: https://forums.frontier.co.uk/showthread.php?t=93710&p=1455408&viewfull=1#post1455408