I read in one russian book about complex vectors. There was formula for product of two complex vectors. I was confused because I didn't find what this formula is. $$ \begin{aligned} \boldsymbol{z}_{1} \boldsymbol{z}_2 &= (\boldsymbol{a}_1 + i\boldsymbol{b}_1) (\boldsymbol{a}_2 + i\boldsymbol{b}_2) \\ &= \bigl[ (\boldsymbol{a}_1, \boldsymbol{a}_2) - (\boldsymbol{b}_1, \boldsymbol{b}_2) \bigr] + i \bigl[ (\boldsymbol{a}_1, \boldsymbol{b}_2) + (\boldsymbol{b}_1, \boldsymbol{a}_2) \bigr] \end{aligned} $$ where $\boldsymbol{a}$ and $\boldsymbol{b}$ are vectors not numbers, $(\boldsymbol{a}_1, \boldsymbol{a}_2)$ and so on is scalar product of two vectors.
What this formula for product of two complex vectors is?