Yeah, this would be a job for javascript (see below)
BUT .. what is the 2nd field used for in your application? If it's for anything important, I wouldn't recommend using javascript for the task. Obviously it can be disabled and time settings vary from one user's locale to another. For important values, it's better to use the server time.
<script type="text/javascript">
function updateDate() {
document.getElementById('otherField').value = new Date();
}
</script>
<form>
<select id="myList">
<option value="1">option 1</option>
<option value="2">option 2</option>
<option value="3">option 3</option>
</select>
<input type="text" id="otherField">
</form>