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.\ $ (Replacing the $k$-th row by itself minus the previous one, for $\ k=2\ldots n,\ $ may further simplify the situation, but I don't know at this time).
Good luck, and I will try to continue too.