Although the physical properties of the 2D and 1D Ising models are quite different, we point out an interesting connection between their complex-temperature phase diagrams. We carry out an exact determination of the complex-temperature phase diagram for the ID Ising model for arbitrary spin s and show that in the u(s) = e(-K/s2) plane (i) it consists of N-c,N-1D = 4s(2) infinite regions separated by an equal number of boundary curves where the free energy is nonanalytic; (ii) these curves extend from the origin to complex infinity, and in both limits are oriented along the angles theta(n) = (1 + 2n)pi/4s(2), for n = 0,...,4s(2) - 1; (iii) of these curves, there are N-c,N-NE,N-1D = N-c,N-NW,N-1D = [s(2)] in the first and second (NE and NW) quadrants; and (iv) there is a boundary curve (line) along the negative real u(s) axis if and only if s is half-integral. We note a close relation between these results and the number of area of zeros protruding into the FM phase in our recent calculation of partition function zeros for the 2D spin a Ising model.