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I want to know if they have a similar type of multiplication for complex numbers.

That is (a+ i b) o (c + i d) = (a c + i b d)

I encounter a situation where such a definition is useful.

In physics I get an expression that looks like the following

(Cos[x1] , Cos[x2] ).A.Transpose[(Cos[y1] , Cos[y2] )] +

i (Sin[x1] , Sin[x2] ).B.Transpose[(Sin[y1] , Sin[y2] )]

where A and B are 2x2 real matrices.

I can express the above expression in the following simpler form, if the complex product "o" as I defined earlier already exists in literature.

(exp[i x1], exp[i x2]) o (A+iB) o Transpose[(exp[i y1], exp[i y2])]

Thanks very much for the help.