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I read they have the same mathematical structure. What does that mean and how far does that sameness go?

If the complex numbers are all ordered pair that obey (a,b)+(c,d)=(a+c,b+d) and (a,b)(c,d)=(ac-bd,ad-bc), can we then equate these ordered pairs with 2-dim vectors? I believe not. What does then (a,b)(c,d)=(ac-bd,ad-bc) mean?

Could someone help?

thanks