Why does Wolfram Alpha say that $\infty^\infty$ results in "complex infinity"?

WA's ComplexInfinity is the same as Mathematica's: it represents a complex "number" which has infinite magnitude but unknown or nonexistent phase. One can use DirectedInfinity to specify the phase of an infinite quantity, if it approaches infinity in a certain direction. The standard Infinity is the special case of phase 0. Note that Infinity is different from Indeterminate (which would be the output of e.g. 0/0).

Some elucidating examples:

• 0/0 returns Indeterminate, since (for instance) the limit may be approached as $\frac{1/n}{1/n}$ or $\frac{2/n}{1/n}$, resulting in two different real numbers.
• 1/0 returns ComplexInfinity, since (for instance) the limit may be approached as $\frac{1}{-1/n}$ or as $\frac{1}{1/n}$, but every possible way of approaching the limit gives an infinite answer.
• Abs[1/0] returns Infinity, since the limit is guaranteed to be infinite and approached along the real line in the positive direction.

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In your particular example, you get ComplexInfinity because the infinite limit may be approached as (e.g.) $n^n$ or as $n^{n+i}$.

TLDR: $\infty$ is not a number, and thus $\infty^\infty$ is meaningless, and Wolfram Alpha is using $\tilde\infty$ to represent something I like to to think of in the sense of "one-point compactification", a topological concept

A.) The symbol $\infty$ is not a number in its own right. It can represent a lot of things, and many different objects can be "infinitely large". Just think of something infinite as something not finite and you are generally off to a good start.

B.) If infinity is not a number, we can't do arithmetic on it that makes sense in all context, and so we most definitely can't exponetiate it meaningfully without some fundamentals first. For example,

1. there are some infinite things where arithmetic is defined (in some sense - check out ordinal numbers),
2. There are some things where basic arithmetic does nothing (i.e. $\infty+1=\infty$)
3. There are some infinite thing where you most definitely cannot do arithmetic in the way you are used to (take, for example, the limit concept of infinity from a basic Calculus class)

C) Wolfram Alpha appears to represent a lot of things as $\tilde\infty$ that are ill-defined according to the real-number system you are used to - for example, according to Wolfram Alpha, $\frac{1}{0}=\tilde\infty$, whereas I would say that $\frac{1}{0}$ is undefined. You could stretch this to say that $\frac{1}{0} = \lim_{x\to 0} \frac 1x = \pm\infty$ in the extended-real number system, but this is starting to push things. To really understand what Wolfram Alpha is doing you must first understand the one-point compactification of $\Bbb C$. See my note at the bottom for links and more details.

Notes:

1. It has been pointed out in the comments that I ought to mention that Mathematica/WA use $\tilde\infty$ to represent an infinite magnitude number with no defined phase. This seemed more software dependent, though I get where the the commentator is coming from. I chose to focus on the real number system the OP is likely accustomed to, and focus on the concept of $\infty^\infty$ itself, not the software's interpretation.

2. $\tilde\infty$ does have meaning in some sense - see for example, this Wikipedia page on "one-point compactification". Just imagine the complex plane as a big sheet where we take the edges and pull them all into one point and call it $\infty$. For more information look at the page on the Riemann Sphere. Similarly, $\infty$ can have a defined meaning in certain contexts outside of the scope of this question - see, for example, the Wikipedia page on the Real Projective Line. In these contexts, one could perhaps consider $\infty$ a number. For more on this, see the Wikipedia page "Projectively Extended Real Line".

An interpretation in $\overline{\mathbb{C}}=\mathbb{C}\cup \{\infty\}$ of $\infty^\infty$ is via limits. For example $$\lim_{z\to 0}\left(\frac{1}{|z|}\right)^{\frac{1}{|z|}}\underbrace{=}_{\text{symbolic equality}}\infty^\infty=\infty.$$ Note. $-\infty$ and $+\infty$ don't belong to $\overline{\mathbb{C}}$.