Is it my alternator????

I have a 2011 825i. Bought it used about a year ago and it has over 11,000 miles, but hasn't given any problems since I have owned it until recently. The other day I was driving it and I stopped and shut it off and then it wouldn't start. It was like it had a really weak battery. So I bought a new battery and installed it and things seemed back to normal for a while. I drove it for most of the day with no problems and all of a sudden it died and then wouldn't start (acting like the battery was dead). Got it back to the shop and parked it for a few day until I had time to mess with it. Tried starting it and it started, but it idled super fast for a minute or two and then the rpm's started surging up and down. I then killed the motor and tried to start it again, and ounce again it was like the battery was dead. Could this be that my alternator is going out?

Thanks for any help or advice.