Let K be a number field and $p$ an odd prime. Let $\mu$ be the Iwasawa $\mu$-invariant of the class group of the cyclotomic $\mathbb{Z}_p$-extension of $K$. If $K$ is abelian over $\mathbb{Q}$ then it is known that $\mu=0$ (Ferrero-Washinton, see Washington 7.5). Iwasawa conjectured that $\mu=0$ for all $K$.
Is something known for the case when $K$ is abelian over an imaginary quadratic field $k$ ?
If I am not mistaken, it proves that mu invariant of the Z_p times Z_p extension is 0 and this was Schneps's thesis. It is unfortunately not enough to show the conjecture of Iwasawa in this case even using the vanishing of anticyclotomic mu invariant proven by Hida.
Edit: In fact, what Schneps proves is that the mu invariant of the Z_p extension in which only one of the primes above p is ramified.
Have you tried looking at Sinnott's paper where he re-proves Ferrero-Washington for $\mathbb{Q}$? It is in Invent. Math, 1984 vol. 75 (2) pp. 273-282.
He proves that to compute the $\mu$-invariant of a function that can be expressed as $\Gamma$-transform of a power series, it is enough to know the $\mu$-invariant of the series. He then applies this to the construction of the $p$-adic $L$-function of Iwasawa where an explicit expression (page 282 and equation (4.3) on page 280) of the paper can be found. Since this ''explicit expression'' comes from the Euler system of cyclotomic units and we now dispose of the Euler system of elliptic units (i.e. we now call it in such a way) for the cyclotomic extension of your imaginary quadratic field, it is plausible that Sinnot's argument applies. But if Coates and Rubin have doubts there must be something tricky behind.
