What is the best of the best program which lets me change the color of the eyes in a picture.
It doesn't matter if it is a free or shareware program.
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Here's a simple tutorial to modify the eye color in Photoshop: http://www.lunacore.com/photoshop/tutorials/tut004.htm
If you don't own photoshop, you can try the same thing in Gimp.
If you are asking the best software regardless size or cost, Adobe Photoshop is always the best choice. If you are looking for something lighter, maybe you should try Gimp.
You might have a look at Portrait Professional if you plan to do a lot of this. It's not super cheap but it does a ton of useful stuff for people photography. The link I provided is the the eye-color changing feature. There are heaps of other features in Portrait Pro. I typically just use Photoshop, but I also post-process only the smallest number of images from each shoot. If I were doing senior portraits or something like that, I'd certainly use this kind of tool.