If you want to check the current page, that is, the default action is the current controller.
$controller = Yii::app()->getController();
$isHome = $controller->action->id === $controller->defaultAction->id ? true : false;
dafeultaction may not always be an "index", it can be changed, so you need to compare it with defaultAction.
And on the home page, if you mean a page with a website defect, you also need to compare your controller with defaultController..
$controller = Yii::app()->getController();
$default_controller = Yii::app()->defaultController;
$isHome = (($controller->id === $default_controller->id) && ($controller->action->id === $controller->defaultAction->id)) ? true : false;
In Yii2:
$controller = Yii::$app->controller;
$default_controller = Yii::$app->defaultRoute;
$isHome = (($controller->id === $default_controller) && ($controller->action->id === $controller->defaultAction)) ? true : false;
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