When a user changes his settings (address, password, picture, etc.) what should be done when he clicks on "send"? Should he be redirected to the
index page, or should he remain on the same settings page? I think both ways are confusing for a user.
Redirecting the user on the home page is confusing as he didn't ask for it when validating.
The most expected behavior is to let him on the setting page, with a success message ('your settings have been updated') and maybe an option to revert the modifications.
The thing to consider here is that the user will have to check if his new settings are correct before moving to another section (which can be going to the index page or any other menu of your site).
First, you definitely should give feedback on successful change, or error in settings, or server error.
After successful change you could redirect user to previous step, i.e. to page from which the user asked to change his settings.
Consider the scenario where the user has the settings page open and another page in another tab, specifically in order to change a setting then go to the other page to see how the setting change effects it. The user might want to go back to the settings to readjust something or undo the setting just set.
If this scenario or one like it is foreseeable in your system then the settings page should stay in place. In general I don't see a reason why the settings page should redirect to something else after a change.