Dear Nate, first, why is $e$ the preferred base for exponentials? Imagine that you have 1 dollar and your bank gives you 100% interest rate. After 1 year, you will have 2 dollars.
Now it offers you to add interests 100 times in a year but the interest is 1% at each moment. How much will you get? You will get
$$ (1+0.01)^{100} \approx 2.704 $$
What if they add you $1/N \times $ 100% at $N$ moments of the year and you send $N$ to infinity? Well, you will have $e\approx 2.71828$ dollars after one year.
In fact, the general exponential - power with the base of $e$ - may be defined by this "repeated small interest" formula as
$$ e^X = \exp(X) = \lim_{N\to \infty} \left(1+\frac {X}{N}\right)^N $$
It only has this simple form if the base is $e$. A more general power may be defined as
$$ Y^X = \exp(X \ln Y) .$$
Here, $\ln$ is the natural logarithm so that $\exp\ln X = X$. If I replaced the base $e$ by another base such as $2$ or $10$, the "repeated small interest" formula above would have to contain $\ln 2$ or $\ln 10$ or other awkward factors at various places. It wouldn't be natural.
So instead of powers $Y^X$ and logarithms with general bases, you should think that in mathematics, only $\exp(X)$ and $\ln(X)$ are really needed, and all the other powers and logarithms may be expressed as composite functions. Also, $\exp(X)$ has the advantage that its derivative is exactly equal to the very same function $\exp(X)$. In particular, the derivative evaluated at $X=0$ is equal to one, very nice and simple. It would be $\ln(Y)$ if you used a different base $Y$ instead of $e$.
Now, what is the exponential of an imaginary exponent? Again, you may write
$$\exp(iX) = \lim_{N\to\infty} \left(1+\frac {iX}{N}\right)^N $$
You multiply $N$ copies of a number that is very close to one. What do you get?
Well, the multiplication by a complex number has the effect of magnifying (or reducing) the plane, and rotating it. In particular, the absolute value of the number $(1+iX/N)$ is essentially one, up to second-order corrections that disappear in the $N\to \infty$ limit. So in the limit, $(1+iX/N)$ is effectively a number whose absolute value equals one.
Multiplying by complex numbers whose absolute value is equal to one looks like a rotation of the complex plane. The angles are preserved - those are some things one should know about the complex numbers. Moreover, it's clear that multiplying by $(1+iX/N)$ is equivalent to the rotation by $X/N$ in radians. If you multiply the same factor $N$ times, you simply get a rotation by $X/N$ in radians.
So the $N$th power of $1+iX/N$, in the limit $N\to\infty$, is the number that you get by rotating $1$ in the counter-clockwise direction by the angle $X$ in radians. Clearly, the answer is
$$ \exp(iX) = \cos (X) + i \sin(X) $$
where the trigonometric functions have arguments in radians, of course. Once again, the mathematically natural unit of an angle is in radians for very similar reasons why the natural base of the powers or exponentials is $e$. Only in radians, it's true that the derivative of $\sin X$ equals $\cos X$ and many other things.
In fact, the previous formula makes it natural to say that $\cos X$ and $\sin X$ are not "independent" functions, either. They may be defined as
$$\cos (X) = \frac 12 ( e^{iX} + e^{-iX} ) $$
$$\sin(X) = \frac{1}{2i} (e^{iX} - e^{-iX} ) $$
You may substitute the last two equations into the previous one or vice versa to check that everything is consistent.
Just to be sure, general complex numbers may also be exponentiated via $\exp(A+iB) = \exp(A)\exp(iB)$ where both factors are known.
In brief: thinking of exponentiation as repeated multiplication is equivalent to the identity $a^{m+n}=a^ma^n$. But focusing on the latter equation instead of the former concept allows us to expand exponentiation beyond the concept of "multiply the base $n$ times", to negative, rational, real, and complex exponents. As you say in the OP question, "the basic rules" allows you to extend past the notion of repeated multiplication for counting numbers. Focusing instead on the equation $a^{m+n}=a^ma^n$ means following the mantra: exponentiation turns additions into multiplications, and rotations are a natural multiplication on the complex plane.
In more detail:
The notion of exponentiation as repeated multiplication is, for natural numbers $n,m$ equivalent to the identity $a^{m+n}=a^ma^n$, because
$$a^{n} = a^{\underbrace{1+\dotsb+1}_{n\text{ times}}}=\underbrace{a\cdot\dotsb\cdot a}_{n\text{ times}}.$$
By focusing on this identity moreso than the notion of "multiplication repeated $n$ times", it also allows us to make sense of exponentiation of naturals, integers, rationals, reals, complexes, even matrices and more, whereas the repeated multiplication notion only makes sense for $n$ natural number, a counting number. We only extend to exponentiation of zero, negatives, and rationals via the above identity (or other similar). Therefore we should view the identity $a^{m+n}=a^ma^n$ not just as a consequence of exponentiation as repeated multiplication, but as a complete and fundamental conceptual replacement. As our conceptual starting point. Exponentiation is, by definition and fundamental conception, the operation that turns addition into multiplication.
As you say in your question, you must extract an understanding of exponentiation of rational, negative, and real exponents via the "basic rules". This identity $a^{m+n}=a^ma^n$ is the most basic of our basic rules, and it will guide us in extracting an understanding imaginary exponents as well.
Now to the matter at hand. On the real line, real numbers act additively by shifting, and multiplicatively by scaling away from zero.
On the complex plane, real numbers act additively by shifting along the real axis (horizontally), imaginary numbers act additively by shifting along the imaginary axis (vertically). Note that orbits of these two actions are orthogonal. Horizontal lines versus vertical.
Real numbers act multiplicatively by stretching away from the origin, while imaginary numbers by rotating $90º$. Note that the orbits of these two actions are also orthogonal. Orbits of scalings are radial lines; orbits of rotations are circles.
Now we have decided that our most fundamental identity of exponentials is $a^{x+y} = a^xa^y$. Exponentiation turns adders into multipliers. It turns real adders (i.e. horizontal shifts) into real multipliers, i.e. scalings away from zero. Horizontal lines into radial lines.
And therefore what must exponentiation transform the orthogonal imaginary shifts, i.e. vertical shifts into? They must transform into those multipliers which are orthogonal to the radial expansions. Which are the rotations. Vertical lines into circles. So exponentiation with an imaginary exponent must be a rotation.
The base of the exponentiation sets the size scale of these stretchings and rotations, and exponentiation with base $e$ does natural rotations in radians, but this picture works with any base of exponentiation, so long as $a>1.$
This intuition is encoded in Euler's identity $e^{i\theta}=\cos\theta + i\sin\theta$. A special case is $e^{i\pi} = -1$, which just says that rotation by $180º$ is the same thing as reflection. This intuitive point of view for understanding Euler's identity is explained in a popular 3Blue1Brown video.
So how do we understand an expression like $x^{\frac{i\pi}{4}}$? Well assuming $x$ is real with $x>0$, since the exponent is imaginary, it's a rotation. How big a rotation? well it depends on the base $x$, and the exponent $\frac{i\pi}{4}$. Computing this magnitude could be thought of as an exercise in operations derived from repeated multiplication, as you set out in your question, but doing so is of limited utility.
Instead we should think of it as an adder $\frac{\pi}{4}$, turned into a vertical shift $\frac{i\pi}{4}$ by the complex unit $i$, and then turned into a rotation by exponentiation with base $x$ (as long as $x$ is real and $x>1.$) The magnitude of $x$ determines the speed of the rotation, or the units.
Best Answer
Consider a real number $A$, and take it to the power $i$. If our system of complex numbers is to be consistent, then $A^i$ must be a complex number; in other words, there must be two real numbers $x$ and $y$, which depend on $A$, such that:
$A^i=x+iy$
Furthermore, we can write $A^{-i}=x-iy$ for the same $x$ and $y$. Hence:
$x^2+y^2=(x+iy)(x-iy)=A^iA^{-i}=A^{i-i}=A^0=1$
We have shown that for any real number $A$, $|A^i|=1$, and therefore $A^i$ corresponds to a complex number which lies some angle $\theta$ along the unit circle.
Now consider the sine and cosine functions for extremely small angles $\epsilon$. A tiny angle $\epsilon$ cuts out a slice of the unit circle, and the curvature of the circumference over this small angle is negligible. We can therefore think of this slice as a right triangle with angle $\epsilon$, and the hypotenuse and adjacent sides are both length one since they correspond to the radius of the unit circle.
Using the formula for the arc length of a circle, it's easy to determine that in the right triangle formed by the small angle approximation, the length of the side opposite to the angle $\epsilon$ is equal to $\epsilon$. We can read off the $(x,y)$ coordinates from this diagram (which are $(cos(\epsilon),sin(\epsilon))$), and therefore we conclude that for very small angles $\epsilon$:
$sin(\epsilon) \approx \epsilon \hspace{10mm} cos(\epsilon) \approx 1$
therefore $cos(\epsilon) + isin(\epsilon) \approx 1+i\epsilon$, and hence for real numbers $A$ which are extremely close to one (so that $lnA$ is small), the complex number $A^i$ lies approximately at an angle $lnA$ along the unit circle, since $A^i=e^{ilnA}\approx 1+i(lnA)$.