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I read this article on understanding imaginary numbers as rotations of real numbers in the complex plane. Having read it, it's easy for me to see how the real number $1$ is simply a point on the real number line, and $1 \cdot i$ is simply a $90^\circ$ rotation of that point.

It's harder for me to see how $1^i$ is also a rotation of that point, as described here.

To investigate, I would like to plot the following set of complex numbers in the complex plane:

$ z(a)\ =\ 1^{ai} $

where a is some real number.

How do I get Wolfram Alpha to understand that I want to plot this function in the complex plane?

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    $\begingroup$ plot f(a) = 1^(asqrt(-1)), possibly including a line to say over what values of a to range. It won't give you anything interesting, try changing the 1 for another number. $\endgroup$
    – Edward Evans
    Apr 9, 2016 at 6:20

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Euler's Formula is as follows;

$$e^{i\theta} = \cos\theta + i \sin \theta.$$

You can think of the number $1$ as $e^{\ln1}$ and hence of $1^i$ as $e^{i\ln1}$. But $\ln 1 = 0,$ so really you have $e^{0i} = e^0 = 1$, as I'm sure you're aware. Even if we look at this by feeding $0$ into Euler's Formula, we have

$$e^{0i} = \cos(0) + i\sin(0) = 1.$$

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