exponentiation - Calculate, simplify and expand exponents with complex numbers

Can we somehow calculate $a^z$ where z is a complex number ?

Does normal exponent rules like :

$$a^b\cdot a^c=a^{b+c}$$

Still work when complex numbers are in the exponent ? For example, do these egalities are true ?

$$2^{4+2i}+2^{3+4i} = 2^{7+6i}$$
$$(2^{4+2i})^{3+4i}=4^{(4+2i)\:\cdot \:(3+4i)}$$