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What's best offshore bank account to open for investment business?

You may look at banks in Latvia and Switzerland as well for this purpose.
 
You are asking a really broad question. What are your preferences? Do you want a bank who charges high processing fees while giving the best quality of service? Or are you after the prices? The list is too long. You might want to break it down by citing some of your personal categories.
 
Just bearer in mind that if it is public investment the company will require a license!