Find the complex number \(z\) satisfying the equation
$$z + \frac { \mathrm { i } z } { z ^ { * } } - 2 = 0$$
where \(z ^ { * }\) denotes the complex conjugate of \(z\). Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
On a single Argand diagram sketch the loci given by the equations \(| z - 2 \mathrm { i } | = 2\) and \(\operatorname { Im } z = 3\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\).
In the first quadrant the two loci intersect at the point \(P\). Find the exact argument of the complex number represented by \(P\).