A method to implement elliptic curve public-key cryptosystem over Z(i) is discussed. The method is in fact the same as the technique that works on Galois fields but here works on Z(i). The curve under Z(i) generates more points as compared to the curve under Galois fields. The security of the system is better due to more number of points on the curve. But there are some difficulties in the new system and through this paper; we try to discuss some of the pitfalls of the new system.