Significance of negative eccentricity For any conic section, the eccentricity is the ratio of the distance to the focus and directrix. If the eccentricity were defined to be negative, would this have any significant meaning/application, or potentially be linked to complex numbers?
It actually doesn't change anything; if you look at the equations for finding the eccentricity, it is the square root of something, and since there are two square roots of a number, it can be both negative or positive.

If you make a plot of a conic section based on the eccentricity, it doesn't change at all. Plot it on desmos graphing calculator using the formula r=$\frac{d}{1-dcos(\theta)}$ where d is the eccentricity (polar coordinates). Nothing happens if you negate the eccentricity.

formula for eccentricity
We've been studying this exact question for all of 5 minutes. We have an answer. No.

This is why: The formula for eccentricity is $c/a$ which is the distance from the center to a foci divided by the distance from the center to a vertex. If eccentricity were negative, either the numerator or denominator would have to be negative. It is IMPOSSIBLE to have a negative distance for $a$ or $c$. Therefore, eccentricity cannot be negative.