Which is more basic, hydrazine or ammonia?
Obligatory link to Evans' pKa table.

Ammonia is more basic than hydrazine, by about one order of magnitude.  This is expected, because the -NH2 group is more electronegative than -H or -CH3.

The second lone pair is not involved in the acid-base reaction, it does not point towards the -NH4+ group.  You can, however, force two lone pairs into close proximity.  This destabilizes the unprotonated form.  The keyword is "proton sponge".

1,8-Bis(dimethylamino)naphthalene has a pKa of 12.3, it's one of the strongest known amine bases.  Compare that to the pKa of aniline, which is something like 4.5.