For a complex number $z\in \mathbb{C}$, the formulas for the real part and the imaginary part the well-known formula:
$$ \Re[z]=\frac{z+\overline{z}}{2}\\ \Im[z]=\frac{z-\overline{z}}{2i} $$
What is the equivalent for complex vectors? Let $\mathbf{z}\in\mathbb{C}^n$, for example
$$ \mathbf{z}=(a+ib,c+id,\dots) $$
An obvious definition is component-by-component aplication: $\Re[\mathbf{z}]=(\Re[a+ib],\Re[c+id],\dots)$.
However, I was hoping that for specific values of $n$, such as $n=2$ or $n=3$ there could be fancier definitions involving the determinant or the cross product, or other products?
The problem I am running into with the component-by-component definition is that studying the invariance group becomes a struggle. For example, it should be relatively easy to show that
$$ f[\mathbf{v}]=\Re[\mathbf{v}]^T\Re[\mathbf{z}] $$
should be the orthogonal invariance group: $O^TO=I$.
$$ \begin{align} f[\mathbf{Mv}]&=\Re[M\mathbf{v}]^T\Re[M\mathbf{z}]\\ &=\Re[\pmatrix{m_{00} & m_{01}\\m_{10}&m_{11}}\pmatrix{a+ib\\c+id}]^T\Re[\pmatrix{m_{00} & m_{01}\\m_{10}&m_{11}}\pmatrix{a+ib\\c+id}]\\ &=\pmatrix{\Re[m_{00}(a+ib)+m_{10}(c+id)]\\\Re[m_{10}(a+ib)+m_{11}(c+id)]}^T\pmatrix{\Re[m_{00}(a+ib)+m_{10}(c+id)]\\\Re[m_{10}(a+ib)+m_{11}(c+id)]}\\ &=\Re[m_{00}(a+ib)+m_{10}(c+id)]^2+\Re[m_{10}(a+ib)+m_{11}(c+id)]^2\\\ &=\frac{1}{4}[(m_{00}(a+ib)+m_{10}(c+id))+\overline{m_{00}(a+ib)+m_{10}(c+id)}]^2+\frac{1}{4}[(m_{10}(a+ib)+m_{11}(c+id))+\overline{m_{10}(a+ib)+m_{11}(c+id)}]^2\\ &=\frac{1}{4}[(m_{00}(a+ib)+m_{10}(c+id))+\overline{m_{00}(a+ib)}+\overline{m_{10}(c+id)}]^2+\frac{1}{4}[(m_{10}(a+ib)+m_{11}(c+id))+\overline{m_{10}(a+ib)}+\overline{m_{11}(c+id)}]^2\\ &=\frac{1}{4}[(m_{00}(a+ib)+m_{10}(c+id))+\overline{(a+ib)}\overline{m_{00}}+\overline{(c+id)}\overline{m_{10}}]^2+\frac{1}{4}[(m_{10}(a+ib)+m_{11}(c+id))+\overline{(a+ib)}\overline{m_{10}}+\overline{(c+id)}\overline{m_{11}}]^2\\ &=\frac{1}{4}[(m_{00}(a+ib)+m_{10}(c+id))+(a-ib)\overline{m_{00}}+(c-id)\overline{m_{10}}]^2+\frac{1}{4}[(m_{10}(a+ib)+m_{11}(c+id))+(a-ib)\overline{m_{10}}+(c-id)\overline{m_{11}}]^2\\ &=\frac{1}{4}[m_{00}a+m_{00}ib+m_{10}c+m_{10}id+a\overline{m_{00}}-ib\overline{m_{00}}+c\overline{m_{10}}-id\overline{m_{10}}]^2+\frac{1}{4}[m_{10}a+m_{10}ib+m_{11}c+m_{11}id+a\overline{m_{10}}-ib\overline{m_{10}}+c\overline{m_{11}}-id\overline{m_{11}}]^2\\ &=\frac{1}{4}[a(m_{00}+\overline{m_{00}})+ib(m_{00}-\overline{m_{00}})+c(m_{10}+\overline{m_{10}})+id(m_{10}-\overline{m_{10}})]^2+\frac{1}{4}[a(m_{10}+\overline{m_{10}})+id(m_{10}-\overline{m_{10}})+c(m_{11}+\overline{m_{11}})+id(m_{11}-\overline{m_{11}})]^2\\ &=\frac{1}{4}[a\Re[m_{00}]+ib\Im[m_{00}]+c\Re[m_{10}]+id\Im[m_{10}]]^2+\frac{1}{4}[a\Re[m_{10}]+id\Im[m_{10}]+c\Re[m_{11}]+id\Im[m_{11}]]^2 \end{align} $$
getting quite verbose...