First, we have to figure out exactly what you meant when you said, "change the background color".
Here's one possibility: If I change the background to a solid, uniform color, would you like it? I don't think so because it would look like the person was floating in space.
Next, suppose I made the new background have two colors: one to simulate the floor and another to simulate a wall behind them and then added appropriate shadows. My guess is that it would be an improvement over the one-color background, but you probably wouldn't like that, either, because even doing all that extra work on the background removes the context that tells the viewer where she was when the photo was taken.
One possibility for "changing the background color" that you might like is to keep the existing background, but make it more orange or more blue. OK, in some cases this might be an improvement. An example of this might be where the subject is primarily lit by a daylight balanced flash, but the background is lit by the warm (i.e., orange) tungsten lights in the room. It might look better if the white balance of the background was similar to tthe white balance of the subject. However, in this case, different color balances just isn't a major issue.
Perhaps, by "change the background color", you meant something different that the options I mentioned above. If so, tell us.
To be honest, all I would do to help this particular photo is to brighten the background so that it could be more easily seen and so her hair wouldn't blend in to the background. Contrast control like this is something that good photographers spend a lot of time on, but that isn't "changing the background color".
HTH,
Tom M