A few things off the top of my head:
Ability to profile is critical. You should be able to profile at various brightness levels without the resulting RGB plot looking like a roller coaster track (see attached JPEG). You can get a pretty good read on this when shopping in a store. Set the colour preset to 6500K then look at an image file with a full range of colours and brightnesses that you are intimately familiar with. (This could be a printer test image saved as an sRGB JPEG that you have on a thumb drive then load into a browser.) There should be no obvious inaccuracies, such as inky shadows or blown highlights or reds that are orangish, etc. Repeat at different brightness levels. IOW: the closer the monitor is out of the box to looking as though you had already profiled it, the better.
Angle of view. You should be able to look at the monitor from different positions without seeing a shift in colour. Sit in front of a monitor you are evaluating; try moving a couple feet to left or right, try standing up part way. You should be able to look at the monitor from at least 45 degrees off-centre from any direction without seeing a noticeable change in colour. This will have an effect on the highlight colour problem you report.
Pixel pitch. The individual dots that make up the image should be small enough at your comfortable viewing distance that you can't distinguish them. This comes down to whether the screen diagonal is appropriate for the native resolution. A 17 inch display with a native 1280x1024 may be appropriate. A 19 inch at 1280x1024 may be too coarse. I'm sure this is as must a matter of eye sight and viewing distance as anything.
Rich OSD (on-screen display) feature set. An inexpensive monitor will probably not have all the adjustments you need. Of course, brightness, contrast, and separate RGB channel adjustments is a beginning, but the ability to adjust per-pixel sharpness is pretty much essential too, as is the ability to move the OSD location.
Orientation rotation is nice. By default monitors are used in landscape orientation; the ability to easily rotate it to portrait orientation is great if you ever need to edit portrait orientation pictures.