= Solution
The trace/norm integrality conditions give the stated congruences for
$\theta=(a+bi+c\sqrt m+di\sqrt m)/2$. Solving them yields these <integral> <bases>:
$$
m\equiv1\pmod4:\quad
1,\ i,\ \frac{1+\sqrt m}{2},\ \frac{i+i\sqrt m}{2},
$$
$$
m\equiv3\pmod4:\quad
1,\ i,\ \frac{i+\sqrt m}{2},\ \frac{1+i\sqrt m}{2}.
$$
The three quadratic subfields have fundamental discriminants $-4,D_m,D_{-m}$. By <discriminant of a biquadratic field>,
$$
D_L=(-4)D_mD_{-m}=16m^2
$$
in both residue classes.
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