https://twitter.com/martinmbauer/status/1763622278128947464?t=YObGhK4ZqjAXrwPaHxB8gw&s=19
Many know Euler's identity, but did you also know that
$|e^{i\pi}| = |\pi^{ie}| = |i^{\pi e}| = 1$
While the claim about $\pi$'s powers feels intriguing to me (for I don't know any better), the claim about $i$'s powers feels outrageous.
$πe = 8.53973422...$
$i^x; x ∈ ℤ$, is one of $\{ 1, i, -1, -i \}$, and that's "if and if only" as far as I know. 8.539... is not an integer so the last identity must be false. Is the quoted tweet just misinformation or am I getting something wrong?
Yes, there is some misinformation here or at least omitted information. The first two terms, $e^{i\pi}$ and $\pi^{ie}$ are well defined by Euler's formula. Both have a complex value with norm $1$ by rewriting them as:
$$ e^{i\theta} = \cos \theta + i \sin \theta $$
for a real number $\theta$ (which is $\theta=\pi$ in the first case and as DarkLordofPhysics and Kevin Dietrich point out in comments, $\theta = e \ln \pi$ in the second case).
The third term $i^{\pi e}$ does not have a conventional definition. Attempts to apply Euler's formula (or similar) result in a multiple valued expression, centering on the multivalued complex logarithm $\ln i$.
The restrictions to make a well-defined function out of the complex logarithm are called branch cuts, and this topic comes up fairly often in related Questions asked here in the past, such as Complex logarithm function.