What you see there is that hyperref colours (cite) links (because you told it so in colorlinks=true, citecolor=CeruleanRef). Normally, when you cite only a certain part of that citation is actually turned into a link, with the numeric style only the number itself, not the brackets are linked. Your natbib fix did not extend the link to the brackets, it just coloured them in.
We can emulate that behaviour by adding \color{CeruleanRef} to the wrapper of the \cite command (I took it from numeric.cbx, if you use numeric-comp.cbx or another style, copy the definition of \cite from there)
\DeclareCiteCommand{\cite}[\color{CeruleanRef}\mkbibbrackets]% <--- this is new
{\usebibmacro{prenote}}
{\usebibmacro{citeindex}%
\usebibmacro{cite}}
{\multicitedelim}
{\usebibmacro{postnote}}
for numeric-comp
\DeclareCiteCommand{\cite}[\color{CeruleanRef}\mkbibbrackets]
{\usebibmacro{cite:init}%
\usebibmacro{prenote}}
{\usebibmacro{citeindex}%
\usebibmacro{cite:comp}}
{}
{\usebibmacro{cite:dump}%
\usebibmacro{postnote}}
Remember: This will not actually create a hyperlink, the text will just look like one. Hyperlinks with ranges that big can only be created in very special cases with a huge amount of work. (With numeric-comp, linking the brackets would not even make sense: Where is the link from the bracket supposed to link to.)
MWE
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[backend=biber, style=numeric-comp]{biblatex}
\usepackage[usenames,dvipsnames,svgnames,table]{xcolor}
\definecolor{CeruleanRef}{RGB}{12,127,172}
\usepackage[colorlinks=true, citecolor=CeruleanRef]{hyperref}
\addbibresource{biblatex-examples.bib}
\DeclareCiteCommand{\cite}[\color{CeruleanRef}\mkbibbrackets]
{\usebibmacro{cite:init}%
\usebibmacro{prenote}}
{\usebibmacro{citeindex}%
\usebibmacro{cite:comp}}
{}
{\usebibmacro{cite:dump}%
\usebibmacro{postnote}}
\begin{document}
\cite{companion,knuth:ct:b,knuth:ct:c} and \cite{baez/article} \cite{aksin}
\printbibliography
\end{document}

One could also use a kludge as suggested in my comment
\makeatletter
\renewcommand*{\bibleftbracket}{\blx@postpunct\textcolor{red}{[}}
\renewcommand*{\bibrightbracket}{\blx@postpunct\textcolor{red}{]}\midsentence}
\makeatother
This does have side-effects though: All opening brackets will be coloured red.
natbibdocumentation, I didn't find "color" there, so I don't know what kind of functionality that is.(Something related and common is to color the reference with hyperref since it's a link, with
– pst Apr 25 '14 at 05:23\usepackage[colorlinks]{hyperref}, but in "[1]" that will color only "1", and not the brackets.)natbib). – moewe Apr 25 '14 at 18:16\bibpunctand have no intention to have hyperlinking, you can go with\makeatletter \renewcommand*{\bibleftbracket}{\blx@postpunct\textcolor{red}{[}} \renewcommand*{\bibrightbracket}{\blx@postpunct\textcolor{red}{]}\midsentence} \makeatother. (Probably this will colour more more things red, if you do not want that we will need a more sophisticated answer.) – moewe Apr 25 '14 at 19:06natbiblinks the citation number (that's why it is turned turquoise-ish) and you colour the brackets to pretend they are linked as well, although they are not. You can get the "pretend look" by using the code in my comment above. But note that this really is not a thing you should be doing.hyperrefcolours links so we know they're there and you make your reader think there is a link even though there is none. – moewe Apr 28 '14 at 18:03[XX]in the bibliography. Is it possible withbiber? I know editingbblbut not too robust sincebiberchanges it. – Sigur Mar 16 '19 at 00:07\formatentry's definition to the first call to\printbibliography. At that time, you can change the colour - example output. – Werner Mar 16 '19 at 04:48\DeclareFieldFormat{labelalphawidth}{\textcolor{red}{\bibleftbracket#1\bibrightbracket}}. My problem now is to change the color of some items, not all. For example, only items from specific.bib. – Sigur Mar 16 '19 at 12:04