As a quick preface to what might be an unnecessary answer, I'd like to just add as a note, I do not believe "...the best complexion of any..." is actually correct English (though I'm not 100% certain). You are more likely to hear "...the best complexion of all..." or "...a better complexion than any..." The word "any" implies that you are making one-to-one comparisons and I do not think it is appropriate to use a superlative with it. Nevertheless, despite my concerns with the provided sentence, using "than" is categorically incorrect.
English has two degrees of comparison: "Comparative" and "Superlative" (note they are not usually capitalized but I'm doing that to draw a distinction between the Comparative form and comparison in general).
The Comparative form is used when making comparison between two things.
English is easier than Japanese.
9 is a more interesting number than 8.
"Easier" and "more interesting" are the comparative forms of "easy" and "interesting" and they are used to compare two things: "English" and "Japanese" in the first example and "9" and "8" in the second.
Comparative adjectives use the word "than" to qualify the comparison.
The Superlative form is usually used making a comparison between one thing to an entire group.
English is the easiest of all languages.
9 is the most interesting number of them all.
"Easiest" and "most interesting" are the superlatives forms of the above mentioned adjectives and they are used to compare "English" and "9" to "all languages (of which English is one)" and "them (numbers) all (of which "9" is one)"
Superlative adjectives use the word "of" to qualify the comparison.
Comparative sentences feel like you are ordering two things. Superlative sentences feel like you are selecting one thing from a group. Thus it is reasonable to use "than" with comparatives (since "than" gives a feeling of sequence or order) and "of" with superlatives (since "of" or perhaps more correctly "out of" can establish a group).