The cult of independence of Central Banker is not new. Central Banker over the world have worked hard for this independence. Whether FED or RBI we do hear how govt is trying to control the chairman/governor. The independence and the prominence attached to the person holding the chair is so overblown, that recently a report stating Dr Rajan refusing to continue for second term, has set panic among the Forex trader.
But asking more specific question-Independence from whom.
Peculiar case is of European Central Bank (ECB). It is not Central Banker to specific country. Not it get public support like its American or Indian counterparts can get. Yanis Varoufakis, former finance minister of Greece, present his analysis in Project-Syndicate. His analysis is colored by the events, he presided over as finance minister of Greece, but he does drives the point home that ECB is vulnerable to creditors( read Germany). and there is so much ECB can do.
Prof. Eichengreen, points out that possibly Euro can never take the place of Dollar, because it is a currency without state. and too many committee are to be consulted for any decision.