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I think you can do that using two rotation axis only (cf. proper Euler angles).

And quaternions are made of two complex numbers.

edit: typo.

>And quaternions are made of two complex numbers.

I don't pretend to grok quaternions (I only use them), but I'm fairly sure a quaternion consists of three imaginary numbers plus a real number.

Sure, but the way they play together can also be expressed by the means of two complex numbers (which makes 4 real numbers in total).

This is not true. You can't factor a quaternion into two complex numbers. Please don't spread misinformation.