To add on to what others have already said...there are a couple of things you need to be thinking about before you can tell if it's the camera or the monitor or both.
The camera AWB does a pretty good job with daylight and normally your pictures and videos look quite OK. When indoors and at night, the white balance can get quite tricky (normally too yellow or red).. So if you are using AWB, in some cases the video color will be very off. Likewise, if you are using a fixed white balance setting, it might look quite wrong if the light source temperature does not match your setting. You might want to read up on how to set custom white balance setting and make sure to set your white balance to match the light source. I wouldn't worry about whether it matches other cameras.
The second thing is whether or not your monitor truly displays the correct color and brightness. Surprisingly, a huge number of monitors fails to diplay proper colors. To put it simply, none of my dozen or so monitors that I have owned display the correct colors out of the box. They all needed correction. You can either manually correct it yourself or get one of the color calibrator kit available for purchase. I now use a Spyder color calibrator but for a long time I just did it manually. The trick is that you have to have a correct color print of a picture and a correct digital file. You manually adjust the color and brightness to match the print then lock it to prevent further changing. A color calibrator will take care of everything for you and it can be had for under $100 and it is the best investment you can make if you're serious about photography and videography.
One very strange thing I have noticed but have no explanation for is that once a digital picture is properly color calibrated, it will look good and "normal" even on a badly miscalibrated monitor but if you are changing the colors based on your wrongly calibrated monitor, it will look awful in almost everything except your own monitor.
