I was reading this question, and, after reading the responses, I felt like I had a much better understanding about how they're just another type of number definition.
Why, then, are they called imaginary? I know there's some arbitrariness to why anything is called anything, but this name in particular probably leads a lot of people to be stuck on the question the poster of the linked questions asked. Almost as if the name itself it trying to encourage us not to trust this crazy "number."
Googling was able to get me that Descartes coined the term, but I couldn't find anything on why, or why the name stuck, even in translation.
Best Answer
A formula was found for expressing solutions of third-degree algebraic equations with real coefficients in terms of addition, subtraction, multiplication, division, and square and cube roots. But in some cases, the number whose square root was to be found was negative. But these paradoxical quantities canceled out, leaving a real number. And now the strange part: when such solutions were substituted into the equation, they checked! That was a reason to pay attention to them.
But they did not arise as quantities in geometry (lengths, areas, etc.) nor in accounting (which is where negative numbers came from), so they were not "real".
(I don't think the concept of "real number" existed until fairly late in the game---some time after all this was done.)