Do I need to buy multiple games to be able to play Diablo 3 with my wife?

Yes. Diablo 3 does not allow offline play. If you want to play with someone else, they must each make it past the log-in screen... which means you will both require a separate copy and associated Diablo 3 licence.

Also of note is that you may only have one Diablo 3 per Battle.net account, so your wife will need to set up an additional Battle.net account to be able to play with you.

If you purchase a physical copy, however, you will be able to share the installation disk, but Diablo 3 does not require a CD to run.

• You do not need a CD to play
• Your Diablo 3 license is tied to your Battlenet account; in other words, though you can use the installation disc to install the basic game on your computer, you will be unable to login to the game servers with your Battlenet account unless you have purchased a license.
• As a result, you and your wife will need to buy two copies to play together. You could share one account, but note that this is against the game's TOS and could result in a permanent account ban(which means you would need lose access to all your characters, progress, and game license, forcing you to buy a new copy).

If you both wish to play the game at the same time, you do need to purchase 2 different games (two different keys). The keys are linked directly with your account. If you log into your account to play and your wife logs into your account - you will be logged out instantly.

You do not technically need to purchase 2 different dvds, but rather 2 different keys. You can do this on Blizzard's Diablo 3 purchase page. (virtual copy)