Positive and negative complex numbers? Can there be such a thing as positive and negative complex numbers? Why or why not? 

What about positive or negative imaginary numbers? 

It seems very tempting to say $+5i$ is a positive number while $-2i$ is a negative number. On an Argand diagram (complex plane) $+5i$ would be represented by a point above the horizontal axis while $-2i$ is a point below the horizontal axis.
[begin crackpot theory]

Mathematicians have committed a great blunder by thinking that a certain imaginary number is $i$ and another is $-i$, when it's really the other way around!!!!

[end crackpot theory]

If one were to take the position above, and try to rewrite all of mathematics consistently with the theory propounded above, the result would be that nothing at all would change.  Which one is called $i$ and which is called $-i$ doesn't matter.

It does in some contexts make sense to pay attention to whether the real part of a complex number is positive.  A Dirichlet series $\displaystyle\sum_{n=1}^\infty\frac{a_n}{n^s}$ converges if the real part of $s$ is greater than the real part of the abscissa of convergence of the series, and diverges if the real part of $s$ is less than the abscissa of convergence.
We may partially order the complex numbers in accord with their real parts so that 
   a+bi &lt; c+di iff a &lt; c. 
Thus the line of imaginary numbers divides positive complex numbers from negative complex numbers.  However, if b and d are not equal, a+bi and c+di are incomparable.  Thus it is no longer true that every positive number is greater than every negative number.  Most positive-negative pairs are incomparable.  This partial ordering has some neat features.  For example, in the Riemann sphere representation of the complex plane, a point at infinity is added to the imaginary line, transforming it into a great circle.  This circle thus divides the Riemann sphere into positive and negative hemispheres.  The imaginary numbers, 0, and infinity, which lie between the hemispheres, are all “neutral”—that is, neither positive nor negative—but all other complex numbers are either positive or negative.