alright just for fun: AMA but only about math, will attempt to speed-explain stuff with as few symbols and equations as possible and see what happens
(esp happy to field questions about stuff that seems basic to you and that you feel like you should've gotten a long time ago!)
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"need" is always a strange word to use in this context. there are lots of bits of math that can basically emulate other bits of math - in this case, you can always replace complex numbers by 2x2 matrices if you want to
the nice thing about complex numbers is that the particular 2x2 matrices they represent come up a lot, for a lot of reasons (really a lot of reasons, it's an insanely long list, quantum mechanics is just one example), and yet they still obey the rules of ordinary arithmetic
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like the basic rules of adding, subtracting, multiplying, and dividing complex numbers are the same as for more familiar numbers (exponentiation is mostly the same but a bit weirder) and it's handy to be able to reuse that experience