The problem I had was changine my Apple ID Alternate Email/Rescue Email address. If that's the same thing you're trying to do, I found the answer at the following link: http://support.apple.com/kb/HT5312 which basically says the following:
You can edit or delete your rescue email address at My Apple ID. To edit your rescue email address:
- Navigate to My Apple ID using your web browser.
- Click "Manage your account"
- When prompted, sign in using your Apple ID and password.
- Click Password & Security
- You'll be asked to answer 2 of your 3 security questions before you can make any modifications. If you are unable to remember your answers, you can choose to send an email to your rescue email to reset your security questions.
I hope that helps.
I'm afraid that's not the case.
Apparently you can't change or delete an Alternate Apple ID. From an Apple support where I asked how to delete an Alternate Apple ID:
"Thank you for sending the screenshot. I've contacted our senior advisor about your concern and as it turns out, it is not possible to edit or remove the Apple ID that you've enter on your account. I apologize for the inconvenience."
Your 'alternate Apple ID' and your 'Rescue address' are not the same thing.
The alternate Apple ID is for example the @me.com version of your @icloud.com address - only people who originally had an @me.com address have this - and as it's the same as the @icloud.com address you can't change it.
The 'rescue address' is a non-Apple address to which password or security question resets can be sent, and is set in the 'Password and Security' section at http://appleid.apple.com
You are evidently both using the same iCloud account, so inevitably you share all the data. iCloud was never designed for mulpitple users. You can't separate firstname.lastname@example.org from email@example.com - they are effectively the same address.
If you want to keep your email separate from your wife's you are going to have to open a completely separate iCloud account; however you won't be able to move your @icloud.com address to it, so it's a choice between changing your email address and putting up with the existing sharing.
Changing your email address is a bit of a hassle as you have to email all your contacts and ask them to use the new one, but this happens quite a lot and most people are used to seeing this. You could also set up a Rule on your wife's iCloud account to forward anything addressed to firstname.lastname@example.org to your new address, so that you would still catch mail from people who hadn't updated and ask them to.
(To set up a Rule go to http://icloud.com and on the Mail page click the cogwheel icon at bottom left: choose 'Rules'.)