[QUOTE=Dr Sardonicus;470514]Taking

a = 10, x = 1, c = 1 + 2*I, m = 11 + 2*I, norm(m) = 125

we obtain

10^(1 + 125*k) + 1 + 2*I == 0 mod (11 + 2*I)

The only integer k for which this holds is k = 0.[/quote]
Oops, I forgot to take Eulerphi(125), which is 100. It doesn't affect the conclusion (which may well be why I forgot). The correct congruence, which also only holds for k = 0, is

10^(1 + 100*k) + 1 + 2*I == 0 mod (11 + 2*I)