A Basic Question[about math]--edited title

[CuriousCanuck]

Forgetting cultural reasons as to why we use Base 10, I am curious. If we could start over, from a mathematical perspective, what base would make the most intuitive sense from a scientific point of view to use?

Base 2?
Base 10?
Base 16?
Base 1,A45,Q3$?
Base something else?

Relating to this question, have there ever been any mathematical (or physics I suppose) questions resolved by converting from Base 10 to another base where answers that were elusive before became more readily apparent?

Does base matter? Do all your base belong to us? (sorry)