If you are only going to have 1 HD weapon make it the 12g/20g pump 20g if your wife/gf is small. I think every home should come with a shotgun. Just like a fire extinguisher. Hope ya never need it but it should be there in case you do.
An AR-15 is a far superior home-defense weapon than a shotgun in my opinion.
A shotgun's recoil can be difficult for some people, especially smaller shooters, whereas an AR-15 has barely more recoil than a .22. Also, AR-15s are generally shorter than shotguns and can be fired one-handed; pump shotguns require two hands. And semi-auto shotguns aren't nearly as reliable as an AR-15; and unless the shooter is extremely consistent, even a pump shotgun is less reliable than an AR.
Sure, a shotgun is usually more incapacitating shot-for-shot, but I can put at least two rounds center mass as fast as I can hit just once with most shotguns. And contrary to popular belief, you do need to aim with a shotgun; at home defense ranges the blast is only going to spread a few inches. Also, an AR is going to have a much higher magazine capacity and will be much faster to reload. In addition, light plastic-tipped .223 varmint rounds will usually punch through soft body armor but will penetrate far less through walls than buckshot, meaning you're less likely to kill your neighbors or family members in another room.
There's nothing wrong with using a shotgun for home defense, but I think an AR-15 is easier to use, more effective (especially against multiple attackers and people wearing soft body armor), and also safer for innocent bystanders because of less penetration through walls. The only real disadvantage is cost.