chrome.downloads.onDeterminingFilenameを使うと最後に保存したフォルダを記憶してくれなくなるがfilenameに絶対パスを指定することでエラーを起こし無理やり機能させる

background.js

chrome.downloads.onChanged.addListener(function(downloadDelta) {
	if (downloadDelta.state && downloadDelta.state.current === "complete") {
		var id = downloadDelta.id;
		setTimeout(function(){
			chrome.downloads.erase({id: id});
		}, 5000);
	}
});
chrome.downloads.onDeterminingFilename.addListener(function(downloadItem, suggest) {
	var dir = "C:\\";
	suggest({filename: dir + downloadItem.filename, conflictAction: "prompt"});
});

manifest.json

{
   "background": {
      "persistent": false,
      "scripts": [ "background.js" ]
   },
   "description": "FilenameConflictAction prompt. AND auto-hide download bar.",
   "manifest_version": 2,
   "name": "DownloadPlus",
   "permissions": [ "downloads" ],
   "version": "1.0"
}