This very short story takes the form of a monologue from the operator of a hyperdimensional private detective service which utilizes complex numbers. The fact that it is delivered "as a onesided conversation" (to quote the author) is the only thing about it that I found appealing or entertaining.
In every other way, it is a rather standard "traveling to other dimensions" story, of which there already many.
It includes some puns, like the title and this passage:
(quoted from Private i)
“We handle your most complex cases.”
I’d had about three drinks when I hit upon the premise and a few more by the time I came up with the slogan. Complex? Get it?

Like this work and this work, it addresses the idea that higher dimensional travel could affect molecular chirality and have medical consequences:
(quoted from Private i)
Mirrorimage humans just wind up confusing the hell out of emergencyroom doctors.

According to the author, it was this passage from which the rest of the story grew:
(quoted from Private i)
The truth is simple, even if the maths is complex. I just made a righthand turn in one of those other dimensions, like the ones the string theorists are always talking about.
I know they’re all curled up. You know what else is all curled up? An exit on the highway.
Just a little twist and you’re in another dimension. Another twist and you’re back on the road again, just headed in the opposite direction. Two wrongs don’t make a right, but two rights make a Uturn. i times i is negative one.

An epilogue to the story explains that the author had difficulty with complex numbers when he encountered them in an undergraduate physics class, but recently had an epiphany:
(quoted from Private i)
Earlier this year, I was thinking back to my attempts to understand complex numbers. A metaphor occurred to me. Lefthand turns and righthand turns are both operations that, if done twice, negate a vehicle’s velocity. I don’t know if this metaphor works for physicists and mathematicians, but it helped me understand how circular motion in a complex plane translated to linear harmonic oscillation.

Indeed, I can say with certainty that this works for physicists and mathematicians. In fact, it isn't a metaphor at all. It is literally what multiplication by i does. If you search the internet for "multiplication by i as rotation" you will find hundreds of websites that explain this clearly, often using pictures like this one:
which I found at https://www2.clarku.edu/~djoyce/complex/mult.html. The point is, if you think of the complex number x+iy as a vector in the plane connecting the point (0,0) to the point (x,y) then multiplying it by i precisely rotates it to the left by 90 degrees. (The author of the story imagines it rotating to the right instead, but that is what multiplication by i does.) And, indeed, doing it twice is the same as multiplying by 1, which reverses the direction. It is a shame that the author's physics professor didn't explain this to the class. It is one of the first things I show my students when we start working with complex numbers.
So, the good news is that the author is absolutely correct, and kudos to him for figuring it out on his own. The bad news is that it is not nearly as original or useful as he thinks because it is already widely known.
The story Private i appeared in the Futures column of the 08 June 2022 issue of Nature. Thanks to Allan Goldberg for bringing it to my attention. 