A modest introductory step only. The following partial algebraization might be useful: the present matrix is given by:

- $\quad a_{kk}\ :=\ 2\cdot(k\ +\ i\cdot k)$
- $\quad a_{km}\ :=\ \min(k\ m)\ +\ \imath\cdot\max(k\ m)$

for $\,\ k\,\ m=1\ldots n\,\ $ and $\,\ k\ne m.\ $ However, we may equivalently consider a matrix obtained from the given one by multiplying all entries by $\ 1-i.\ $ We obtain a matrix $\ (b_{mk})\ $ as follows:

 - $\quad b_{kk}\,\ :=\,\ 4\cdot k$
 - $\quad b_{km}\,\ :=\,\ (k+m)\ +\ \imath\cdot|k-m|$

for $\,\ k\,\ m=1\ldots n\,\ $ and $\,\ k\ne m$.

>*Good luck, and I will try to continue too.*