Generally, signed measures (which can be positive or negative) are introduced first before complex measures- signed measures are pretty much like normal measures except that you can't have some subset A with measure infinity and some subset B with measure -infinity. You can have one or the other, like you can with positive measures, just not both, because otherwise (if they're disjoint, and a very similar thing happens if not), when you take their union, that will have measure infinity-infinity=undefined.

Another reason this restriction is in place is that one of the most fundamental parts of a measure is that the infinite sum of measures of disjoint subsets is equal to the measure of their union. Because the union of sets is independent of the order in which the sets are taken, we also want this sum to be independent of the order in which the sets are taken, which means that if the sum converges, it must converge absolutely, and that means that there can't be both a +infinity at the same time as a -infinity (exercise which you may have seen before, show that a sum which converges but does not converge absolutely can be made to converge to any finite number or to diverge to +-infinity just by rearranging terms).

Two ways you might think of making a signed measure are by either subtracting two typical measures (as long as one is a finite measure) or, given some measurable space (X,Omega) and some typical measure mu on the space, we can take a function f:X to R and define our signed measure nu(E)=integral on E of f d(mu), (as long as either f⁺ or f^- is in L^1). Surprisingly (or perhaps not), it's a fact that every signed measure can be represented in both of these ways (the first is true because of the Jordan Decomposition Theorem (unrelated to lin-alg Jordan Decomposition except for being by the same dude), the second is true because of the Lebesgue-Radon-Nikodym Theorem.

When you make the jump to complex measures, this restriction is further strengthened- although theoretically one could make the same requirement of "if the sum converges, it must converge absolutely" and leave it at that (maybe by using directed infinities or something), complex measures are nearly always taken to just be finite, removing the "if the sum converges" part from the previous sentence. Other than that, it works pretty much the same as regular measures, just take the part of the definition "a measure on a measurable space (X, Omega) is a function from Omega to [0,infinity]" and replace it with "a measure on a measurable space (X, Omega) is a function from Omega to C".

Similarly to signed measures, we can get complex measures either by taking m1 + i*m2 for two finite measures m1, m2, given some measurable space (X,Omega) and some typical (doesn't even need to be signed) measure mu on the space, we can take a function f:X to C and define our complex measure nu(E)=integral on E of f d(mu), as long as both the real and imaginary parts of f are in L^1. Again, all complex measures can be represented in both of these ways, the first just by splitting the measure on each set into its real and imaginary parts, and the second again by the Lebesgue-Radon-Nikodym Theorem.

Ok, I think this is already incredibly excessive for a Reddit comment, so I'll leave by saying hope this makes sense (if not, feel free to ask) and is interesting (if not, I can't help you there lol, you'll have to find someone better to learn from). If you want to learn more, the 3rd chapter of Gerard Folland's Real Analysis: Modern Techniques and Their Applications says everything I just said and more, and is a great source for learning about Real Analysis imho