- If you proceed step by step, passing from '2 power 4' via '2 power 3/4' to '2 power pi' and even to '2 power i' (exponentiation with pure imaginary exponent), you will probably stop at each new kind of abstraction. Each time you will try to ask your power of imagination for plausibility of the operation and the result. E.g., try to visualize
e**(i*pi)= -1.
The view changes when you consider the whole exponential function in one, namely exp: Real numbers ---> Real numbers, defined as exp(x):= e power x. Apparently that's a continous and even differentiable function defined for all real arguments. But even more: Without any problem you can extend the domain of definition to the set of complex numbers, e.g. by considering the power series expansion of the exponential function. Hence, what one considers plausible, depends on the level one has already attained in the field in question.
I do not consider complex numbers artifically defined. And for me it is not necessary to legitimate complex exponentiation by reduction via exp(iz) = cos z + i sin z to trigonometric functions. I consider it a deep insight of Gauss that solely by introducing a single imaginary number 'i', complex numbers derive as z = x + iy and each polynomial gets as many zeros as the degree of the polynomial indicates.
Mathematics is not self-evident. Because evidence always depends on the degree to which one is familiar with the subject and on the depth one has penetrated the given problem. Why mathematics fits to solve real-world problems, is still an open question, see Wigner's paper quoted in Alexis' comment.