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The color of a budgies eyes can vary depending on the variety or mutation that they are as well as their age. The color can range from red, ruby, plum and black. 

Here are the mutations that have each eye color, as well as picture examples of each color compared to the others for easy reference.
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Black eye with white iris
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Black eye with developing white iris
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Black eye with no iris
Most mutations have black eyes from the time they hatch through to maturity. Most will also develop a white iris ring as they age with the exception of recessive pied, dark eyed clear and english fallow.

Below are chicks ranging from 1 to 12 days in age to show what a normal black eyed chick looks like.
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1 Day Old Normal Chick
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8 Day Old Normal Chick
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12 Day Old Greywing Chick
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Red/ruby eye with white iris
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Red/ruby eye with developing white iris
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Red/ruby eye with no iris
Red or ruby eyes are a result of the melanin or pigment being removed from the budgies eyes. Budgies with red or ruby eyes are just as healthy as budgies with black eyes so long as they are well bred from a good breeder as with black eyed budgies.
Chicks that hatch with red eyes keep the color for their whole live, and they are easy to distinguish from the moment they hatch by the light pink color of their eyes which can be seen through the skin. Below are some examples of red/ruby eyed chicks at various days in age.
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2 day old albino chick
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6 day old albino chick
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9 day old albino chick
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2 Day Old Cinnamon Chick
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6 Day Old Df Spangle Chick
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11 Day Old Cinnamon Chick
A number of mutations have plum eyed chicks for the first week after they hatch. After this the eyes begin to darken and the plum color is barely noticeable in most lighting. It could be easily missed if you do not have a normal black eyed chick to compare to, or are familiar with what a plum eyed chick looks like.
Cute Little Birdies Aviary