Ca(Fe1-xRhx)(2)As-2 undergoes successive phase transitions with increasing Rh doping in the T = 0 limit. The antiferromagnetic-metal phase with orthorhombic structure at 0.00 <= x <= 0.020 is driven to a superconducting phase with uncollapsed-tetragonal (ucT) structure at 0.020 <= x <= 0.024; a non-superconducting collapsed-tetragonal (cT) phase takes over at x >= 0.024. The breakdown of Fermi-liquid transport is observed in the ucT phase above T-c. In the adjacent cT phase, Fermi-liquid transport is restored along with a disappearance of superconductivity. This interplay of superconductivity and Fermi-liquid transport suggests the essential role of magnetic fluctuations in the emergence of superconductivity in doped CaFe2As2.