Can you have addition polymerisation with monomers other than alkenes?
$\ce{N2H2}$ is a very unstable molecule, prone to disproportionation into hydrazine and nitrogen. $\ce{N-N}$ bonds except one in $\ce{N2}$ molecule are unstable and generally unfavorable. So, no there are no polymers with nitrogen as a sole component of the main chain. 

Several classes of compounds of 3-row elements demonstrate tendency for oligomerization, namely $\ce{(HP)_{n}}$ derivatives usually exist as a ring. It is possible to obtain metastable at room temperature plastic sulfur, containing linear sulfur chains terminated by free radicals. 

However, generally when in search of easily polymerizing molecule that produce main chain containing elements other than carbon, one usually have to look into compounds producing chains with two elements. The most common and known one is formaldehyde $\ce{CH2=O}$, producing $\ce{HO-(CH2-O)_{n}-H}$ chains. A less known, but interesting example is hexachlorotriphosphazene, the source material for production of Polyphosphazenes.

In  general, metastable double bonds are rare for elements other than carbon (sometimes with nitrogen or oxygen partners), so usually reactions intended to produce such bonds, say, $\ce{Si=Si}$ usually end with olygomer instead, typically of ring structure. Moreover, single homoelemental bonds are usually much weaker for elements other than CHNOB, so typically when a chains containing other elements is occurred, it is usually hetero-atomic chain, produced by means other than olygomerization of double-bond containing monomer. For example, an extremely wide-used family of polymers (polysiloxanes, or silicones) is produce by hydrolisys of dialkyldicholorsilanes $\ce{R2SiCl2}$.