The proposition of the posting can't be true in general, I'm afraid: e.g., the isometry group of $\mathbb{C}^2$ has (real) dimension $4+6=10$, while $\dim U(2)+\dim O(2)+4=9$. However, the isometries of $\mathbb{R}^n$ (and in particular, of $\mathbb{C}^n$) are easy to describe.

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a map such that $d(x,y)=d(f(x,f(y))$ for all $x,y\in\mathbb{R}^n$ where $d$ is the metric induced by a positive definite quadratic form $(\cdot,\cdot)$.

Suppose $f$ preserves the origin. Then for all $x,y\in\mathbb{R}^n$ of unit length such that $(x,y)=0$ we have $(f(x),f(y))=0$, since $f(x)$ and $f(y))$ are of unit length and the distance between them is $\sqrt{2}$.

Now choose an orthogonal frame $e_1,\ldots,e_n$ of $\mathbb{R}^n$ and take a composition $h=g\circ f$ such that $g$ is of the form $x\mapsto Ax+b$ for some $A\in O(n),b\in\mathbb{R}^n$ and $h$ preserves the origin and any $e_i$. If $x=\sum a_i x_i\in\mathbb{R}^n$ and we know $d=d(x,0)$ and all $d_id(x,e_i)$, then we can recover $a_i$ as $a_i=\frac{1}{2}(d^2-d_i^2+1)$. So the map $h$ is in fact the identity, which means that $f$ is a composition of a translation and an orthogonal map.

The question of the posting is the case when $\mathbb{R}^n=\mathbb{C}^{n/2}$ and $(\cdot,\cdot)$ is the real part of the standard hermitian form.

The proposition of the posting can't be true in general, I'm afraid: e.g., the isometry group of $\mathbb{C}^2$ has (real) dimension $4+6=10$, while $\dim U(2)+\dim O(2)+4=9$. However, the isometries of $\mathbb{R}^n$ (and in particular, of $\mathbb{C}^n$) are easy to describe.

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a map such that $d(x,y)=d(f(x,f(y))$ for all $x,y\in\mathbb{R}^n$ where $d$ is the metric induced by a positive definite quadratic form $(\cdot,\cdot)$.

Suppose $f$ preserves the origin. Then for all $x,y\in\mathbb{R}^n$ of unit length such that $(x,y)=0$ we have $(f(x),f(y))=0$, since $f(x)$ and $f(y))$ are of unit length and the distance between them is $\sqrt{2}$.

Now choose an orthogonal frame $e_1,\ldots,e_n$ of $\mathbb{R}^n$ and take a composition $h=g\circ f$ such that $g$ is of the form $x\mapsto Ax+b$ for some $A\in O(n),b\in\mathbb{R}^n$ and $h$ preserves the origin and any $e_i$. If $x=\sum a_i x_i\in\mathbb{R}^n$ and we know $d=d(x,0)$ and all $d_i=d(x,e_i)$, then we can recover $a_i$ as $a_i=\frac{1}{2}(d^2-d_i^2+1)$. So the map $h$ is in fact the identity, which means that $f$ is a composition of a translation and an orthogonal map.

The question of the posting is the case when $\mathbb{R}^n=\mathbb{C}^{n/2}$ and $(\cdot,\cdot)$ is the real part of the standard hermitian form.