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 $d_id(x,e_i)$, 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)$is the real part of the standard hermitian form. 1 The proposition of the posting can't be true in general, I'm afraid: e.g., the isometry group of$\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.