A "unicode complete" font is more of a novelty than anything else. For real work, pick a collection of fonts that cover the space you need. Good Arabic or Chinese fonts are not going to be great Latin fonts. For that matter, good Chinese fonts are not good Japanese fonts, even if coverage is good enough. In fact, the order of precedence that these fonts have will depend on which language is the primary language. If you're typesetting Japanese with Latin characters, then you can just pick a nice Mincho font, but Mincho has unnatural spacing if your audience has native English readers. (So if you quote Japanese in an English text, pick a Latin font and use Mincho for the Japanese; but if you quote English in a Japanese text, you can use Mincho for everything.)

It's hard enough just to find a good Latin font that also has a good set of mathematical symbols. "Computer Modern" of course, and "Times", but these sets are rare.