  We investigated the effect of electron and hole doping on the high-field
low-temperature superconducting state in CeCoIn$_5$ by measuring specific heat
of CeCo(In$_{\rm 1-x}$M$_{\rm x}$)$_5$ with M=Sn, Cd and Hg and $x$ up to 0.33%
at temperatures down to 0.1\,K and fields up to 14\,T. Although both Cd- and
Hg-doping (hole-doping) suppresses the zero-field $T_c$ monotonically, $H_{c2}$
increases with small amounts of doping and has a maximum around $x$=0.2%
(M=Cd). On the other hand, with Sn-doping (electron-doping) both zero-field
$T_c$ and $H_{c2}$ decrease monotonically. The critical temperature for the
high-field low-temperature superconducting state (so called {\it Q}-state)
correlates with $H_{c2}$ and $T_c$, which we interpret in support of the
superconducting origin of this state.
