u can do it by two ways :--
1) by converting base 4 into decimal adding them and than converting them back to base 4.
=(2.3)4 + (1.2)4
= (2* 40+ 3*1/4) + (1* 40 + 2+1/4) =
=3 + 5/4 ( here u can minimise the values as)
=3+(1 + 1/4)
=4+1/4 u can directly convert it to base 4 now (no need to divide)
= (10.1)4
2) as every number has base 4 so adding them both and if we get an overflow we take the values as xmod4 and extra value as carry.
( 2 . 3)
+ (1 . 2)
10 .1
( for decimal values 3+2=5 and 5 mod 4=1 carry =1
for units place 2+1+1 = 4 and 4mod 4 = 0 carry =1)