I personally don’t really care about character customization, but If I am given the opportunity to customize a character, I do have a few rules that I’ve always gone by:
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If they have a fixed name (or even a suggested name), I don’t touch them. A predetermined name let’s me know that the author had a character in mind (at least somewhat), and I don’t want to alter that.
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If there’s only limited CC, I don’t touch them. Again, that tells me the author had a certain type of character in mind, and I’m not changing that.
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If given full CC, I always attempt to make my characters vastly different. The only time that I actually create the same character twice, is if I’m reading a story that has a sequel.
And I think I’ve only ever read one story that had visual customization, so I’m definitely more familiar with normal customization.
So in a nutshell: Normal customization. Full CC or no CC.