Solve the equation \(z ^ { 2 } - 2 p \mathrm { i } z - q = 0\), where \(p\) and \(q\) are real constants.
In an Argand diagram with origin \(O\), the roots of this equation are represented by the distinct points \(A\) and \(B\).
Given that \(A\) and \(B\) lie on the imaginary axis, find a relation between \(p\) and \(q\).
Given instead that triangle \(O A B\) is equilateral, express \(q\) in terms of \(p\).