The spaces $ω^p_0$, $ω^p$ and $ω^p_∞$ can be considered the sets of all sequences that are strongly summable to zero, strongly summable and bounded, by the Cesaro method of order 1 with index p. Here we define the sets of sequences which are related to strong Cesàro summability over the non-Newtonian complex field by using two generator functions. Also we determine the β-duals of the new spaces and characterize matrix transformations on them into the sets of *-bounded, *-convergent and *-null sequences of non-Newtonian complex numbers.