We propose a unified superfield formulation of N=4 off-shell supermultiplets in one spacetime dimension using the standard N=4 superspace. The main idea of our approach is a gluing together of two linear supermultiplets along their fermions. The functions defining such a gluing obey a system of equations. Each solution of this system provides a new supermultiplet, linear or nonlinear, modulo equivalence transformations. In such a way we reproduce all known linear and nonlinear N=4, d=1 supermultiplets and propose some new ones. Particularly interesting is an explicit construction of nonlinear N=4 hypermultiplets.