The generalization of the phenomena you see is genus theory. If $K = \mathbf{Q}(\sqrt{-d})$ and $H = K(j_d)$ then $H$ contains the Genus field $G$.
If $d = \prod_{i=1}^n p_i$ is squarefree (and odd for convenience's sake) then $G = K(\sqrt{p_i^*})$ where $p_i^* = (-1)^{(p-1)/2}$. -1)^{(p_i-1)/2}p_i$. In particular$p_i^* = p_i$if and only if$p_i \equiv 1\bmod 4$. Therefore if$d \equiv 1\bmod 4$then$d = \prod_i p_i^*$. Therefore if$d \equiv 1\bmod 4$then$G$and thus$H$contains$K(\sqrt d) = K(i)$. In general, if$d_1$and$d_2$have lots of prime factors in common, then the genus fields of their corresponding imaginary quadratic fields will also have large intersections. Outside of the genus field, I don't know of any studies into the intersections of Hilbert Class Fields. The standard reference for this is Cox's wonderful book "Primes of the form$x^2 + ny^2$." 1 The generalization of the phenomena you see is genus theory. If$K = \mathbf{Q}(\sqrt{-d})$and$H = K(j_d)$then$H$contains the Genus field$G$. If$d = \prod_{i=1}^n p_i$is squarefree (and odd for convenience's sake) then$G = K(\sqrt{p_i^*})$where$p_i^* = (-1)^{(p-1)/2}$. In particular$p_i^* = p_i$if and only if$p_i \equiv 1\bmod 4$. Therefore if$d \equiv 1\bmod 4$then$d = \prod_i p_i^*$. Therefore if$d \equiv 1\bmod 4$then$G$and thus$H$contains$K(\sqrt d) = K(i)$. In general, if$d_1$and$d_2$have lots of prime factors in common, then the genus fields of their corresponding imaginary quadratic fields will also have large intersections. Outside of the genus field, I don't know of any studies into the intersections of Hilbert Class Fields. The standard reference for this is Cox's wonderful book "Primes of the form$x^2 + ny^2\$."