As per the inquiry,

We have,

\[\left| z \right|\text{ }=\text{ }z\text{ }+\text{ }1\text{ }+\text{ }2i\]

Subbing z = x + iy, we get,

\[\begin{array}{*{35}{l}}

\Rightarrow \left| x\text{ }+\text{ }iy \right|\text{ }=\text{ }x\text{ }+\text{ }iy\text{ }+\text{ }1\text{ }+\text{ }2i  \\

~  \\

\end{array}\]

We realize that,

\[\left| z \right|\text{ }=\text{ }\surd \left( x^2\text{ }+\text{ }y^2 \right)\]

\[\surd \left( x^2\text{ }+\text{ }y^2 \right)\text{ }=\text{ }\left( x\text{ }+\text{ }1 \right)\text{ }+\text{ }i\left( y\text{ }+\text{ }2 \right)\]

Looking at genuine and nonexistent parts,

We get,

\[\surd \left( x^2\text{ }+\text{ }y^2 \right)\text{ }=\text{ }\left( x\text{ }+\text{ }1 \right)\]

Also, 0 = y + 2

⇒ y = – 2

Subbing the worth of y in \[\surd \left( x^2\text{ }+\text{ }y^2 \right)\text{ }=\text{ }\left( x\text{ }+\text{ }1 \right),\]

We get,

\[\Rightarrow x^2\text{ }+\text{ }\left( -\text{ }2 \right)^2\text{ }=\text{ }\left( x\text{ }+\text{ }1 \right)^2\]

\[\Rightarrow x^2\text{ }+\text{ }4\text{ }=\text{ }x^2\text{ }+\text{ }2x\text{ }+\text{ }1\]

Consequently, x = 3/2

Consequently, z = x + iy

= 3/2 – 2i