A weird idea on definition of completing a metric space Please note the definition below, captured from Page 102, Real Analysis, Carothers, 1ed:

Completions
Completeness is a central theme in this book; it will return frequently. It may comfort you to know that every metric space can be "completed." In effect, this means that by tacking on a few "missing" limit points we can make an incomplete space complete. While the approach that we will take may not suggest anything so simple as adding a few points here and there, it is nevertheless the picture to bear in mind. In time, all will be made clear!
First, a definition. A metric space $(\hat M,\hat d)$ is called a completion for $(M,d)$ if
(i) $(\hat M,\hat d)$ is complete, and
(ii) $(M,d)$ is isometric to a dense subset of $(\hat M,\hat d)$.

Let $d(x,y)=|x-y|$ and $\hat d = d$. Then if $M=(0,1)$ and $\hat M$ is $[4,5]$, $\hat M$ is a completion of $M$ because $M$ is isometric to $(4,5)$ which is a dense set of $[4,5]$, the complete metric space. It is really interesting that intersection of $\hat M$ and $M$ is empty, say to complete a metric space, it is not necessary for us to use those old elements in $M$, which is a little ridiculous.
Is there any problem about the idea above?
 A: In my opinion, the right way of thinking about completion is not by looking at the completion $(\hat{M}, \hat{d})$ but looking at the completion and the isometry $(\hat{M}, \hat{d}, f)$.
A completion of $(M,d)$ is actually a triple $(\hat{M}, \hat{d}, f)$, where $(\hat{M}, \hat{d})$ is complete, $f(M)$ is dense in $\hat{M}$ and 
$$f: (M,d) \to (f(M), \hat{d}) \, \mbox{ is an isometry} \,.$$
Any confusion will appear only if we ignore/forget $f$, but this is a part of the completion...
A: It really does not make sense to talk about a completion in terms of using the "old elements" without some choice of an ambient complete space. Sure, if we think of $(0,1)$ as living inside $\mathbb{R}$ then there is in some sense a canonical choice for the completion: $[0,1]$. But what if I give you an abstract metric space? I can't just "add new elements to the old elements", because my metric space only has the elements I started with. I need to move to a new set, and speaking philosophically that requires me to replace my old set with an isometric copy living in a larger space where I can add points.
