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I am working on a project that needs vector math and complex numbers.
I am looking a for a good open source API that supports C/C++ and hopefully has decent performance.
I can write these functions myself but it will be ugly and slow.
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I regularly use Sony Vector Math library bundled with bullet physics. It's fast has many linear algebra algorithms.
For complex numbers you can simply use std::complex
You can use the C++ complex numbers library
Also, Boost provides linear Algebra package
I haven't used any of these packages personally, but head over to Object-Oriented Numerics for a great selection of available libraries
Eigen is pretty much standard in research and should offer the stuff you need as well.