The job of a monitor is to accurately show you the contents of an image file. Bit depth matters. You can see the difference.
Beware of some 'enhanced' monitors especially some sold by Apple, they alter the image to give you a stunning image, fine for viewing a film, but that's not what you want when editing images, stills or video, you want accuracy then.
The higher the bit depth the more accurately the data is reproduced. Every bit you add doubles the amount of data you can see. It's important because dark shadows block out to black and bright pixels burn out to white more readily the lower the bit depth.
Jpegs only require an 8bit monitor, but every photographer I know shoots Raw, the difference is visible and you can edit drastically without destroying the image.
Most DSLR cameras are 12 or 14bit when shooting Raw, the cheapest 12bit monitor, which is good in other respects too, is the Dell U2140, you would have to double the price to get a decent 14bit monitor, and sell your car to get a 16bit monitor.
That's not the end of the story though, to get a really accurate view of your image files so the view on your monitor is actually what is in the file data, you have to calibrate your monitor too using a colourimeter such as ColorMunki or Spyder. The ColorMunki has the advantage of it can calibrate your printer and projector too.
Only when all this is in place can you send your images off to be printed and they will closely match what you see on screen, it's never perfect as screens are additive colour and prints are reflective colour, print are usually slightly dimmer than the screen. Even that can be allowed for if you view the image with the colour profile the print house uses, all the Pro print shops make their colour profile available and programs such as Photoshop can show you the image viewed through it.
I bet you thought it was a simple question LOL.
Addition:- you only need a 10bit monitor to show Adobe RGB colour space, 8bit to show sRGB (the same colour space that Jpeg uses) and 16bit to show all the colours available in Pro Colour Space.
Chris