How to convert wave from real to complex and vice versa?
Rather than saying the questions you are asking are non-sensical, I will say they reveal a lack of understanding of complex numbers.  There is lots of material to be had with a few simple searches.  I would also recommend that you read my blog article The Exponential Nature of the Complex Unit Circle which gives an explanation of Euler's equation which is absolutely essential to understand in order to comprehend the meaning of DFT bin values.

For your first question:  Technically, a real number is a complex number with a zero imaginary component.  So setting the real part equal to your values and the imaginary part to zero is the correct method.  When you multiplied it by $ i^{0.4} $ it should have rotated each value a tenth of a cycle.  Thus the amplitude stays the same, but the phase shifts by $\pi/5$ radians.

For your second question:  A complex number consists of two values.  A real number consists of one.  Therefore converting a complex number to a real number is a non-sensical proposition.  However, you can calculate the magnitude, which is a real number.  You can also ask what the real part is, which is also a real number.  The imaginary part is a real number multiplied by $i$.

A real tone is a sinusoidal, flat as a pancake.  A complex tone is a spiral, like a slinky.  The definitions I use in my blog articles are:

$$ S_n = A \cos( \alpha n + \phi ) $$

For a real tone, and:

$$ S_n = A e^{i (\alpha n + \phi)} $$

For a complex one.

Notice that a real valued tone is just the average of two complex tones:

$$ \frac{ A e^{i (\alpha n + \phi)} + A e^{-i (\alpha n + \phi)} }{2} = A \frac{ e^{i (\alpha n + \phi)} + e^{-i (\alpha n + \phi)} }{2} = A \cos( \alpha n + \phi )$$

Because

$$ \cos( \theta ) = \frac{ e^{i\theta} + e^{-i\theta} }{2} $$

Hope this helps.

Ced