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Calculators, probably in the 1980's (I know for a fact the HP 48 calculator circa 1992 handled complex and imaginary numbers) helped people perform calculations with complex numbers, without having to figure conjugates, angles, etc. on paper.
Complex number computing was long before that.
I know that FORTRAN (developed by IBM and released in 1957) could handle complex number calculations. FORTRAN was designed specifically for scientific and engineering calculations. Check out the Wikipedia article.
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