Curious but true!
First of all, a “negative imaginary number” is not an imaginary number with a smug and cynical attitude. (That one was a lame joke on my part.)
Secondly, I find that it is very hard to grasp a negative imaginary number without contextualizing it in some real world application of physics, such as the electrical power transmission situation which @Patrick mentioned. For example: an “i” in a mathematical expression which is related to power losses can be viewed as a negative.
Also, it often helps to conceptualize even imaginary numbers on a traditional “number line”, just like the one that is sometimes used in elementary school arithmetic classes. Yes, trying to put “i” and complex number multiples of it onto that number line is challenging but at the very least we can think of such numbers as positioned to the left of zero and therefore negative.
I remember the first time I learned about imaginary numbers. I thought they were the coolest numbers I’d ever encountered. (Even more so than primes.) The fact that they “don’t mean anything” on their own and yet they arise when working out the algebraic equations of various physical processes—and then tend to “disappear” when one gets to the last step and the “real world” solution of a problem----seemed absolutely glorious to me!
As a young person learning about imaginary numbers, I ended up simply accepting “i” by faith because it was so useful in making sense of the real world. It is an almost “magical” thing which one can’t see and yet must be there for physics to make sense. I don’t know any other way to describe it.
I realize that “transcendental” in terms of transcendental numbers is yet another category of numbers (which can involve “i”) but I’ve always felt that “i” and imaginary numbers are in a traditional religious studies academic sense almost (but not really) “transcendent” in how they relate to the real world: somehow “in” the real world but existing independently in some grand way. They are just plain cool. And imaginary numbers tend to pop up just about everywhere in physics.