Google Apps Script bounded to spreadsheet project that logs the timestamp and user email next to the edited cell for an specified range. It assumens that the spreadsheet has only one sheet.

Code.gs

/**
 * Log the timestamp and user email next to the edited cell
 * @param {Event} e Edit event object
 *
 */
function logWhenWho(e){
  
  // Get timestamp ASAP for the best precision
  var timestamp = new Date();
  
  // Validate trigger
  var triggerId = PropertiesService.getUserProperties()
                    .getProperty('timestamp_trigger');
  SpreadsheetApp.getActive().toast('triggerUid: ' + e.triggerUid);
  if(e.triggerUid !== triggerId) {
    //console.log('Terminated: Invalid trigger');
    return;
  }
  
  // Validate edited range
  
  /* Column of monitored range */
  var monColumn = 3;
  
  /* Starting row of monitored range */
  var startRow = 2;

  var column = e.range.getColumn();
  var row = e.range.getRow();
  if(column !== monColumn || row < startRow) {
    //console.log('Terminated: invalid range');
    return;
  }
  // Main
  var user = Session.getEffectiveUser();
  var whenWho = [[
    /* When */
    timestamp, 
    /* Who */
     user.getEmail()
  ]];
  e.range.offset(0,1,1,2).setValues(whenWho); 
  //console.log('Terminated: whenWho added');
}

/**
 * Create spreadsheet on edit strigger
 *
 */
function createTrigger() {
  var trigger = ScriptApp.newTrigger('logWhenWho')
    .forSpreadsheet(SpreadsheetApp.getActive())
    .onEdit()
    .create();
  var id = trigger.getUniqueId();
  PropertiesService.getUserProperties()
    .setProperty('timestamp_trigger', id);
}
/**
 * Add a menu to manage triggers
 *
 */
function onOpen(e){
  SpreadsheetApp.getUi()
    .createMenu('Log When-Who')
    .addItem('Add trigger', 'createTrigger')
    .addToUi();
}

Thanks for the code.

How can we add hour and minute and also arrange the target cell to write the date and time?