There are a lot of services offering the same, even offering archiving but they often need registration.
Now with jetwick it is easy to search any account you like . E.g. try my account. To do the same for your account go to Jetwick, click ‘login’, allow jetwick access to your account (you can revoke it at any time and we won’t misuse it or even post tweets etc) and then click “grab tweets”.Then you will see something like
After this procedure you can search the whole history of the tweets easily.
Why do you want to grab tweets from other users? With that you can easily see about which topic a user tweets, on the right side. Again see “Words related to your query” of my account:
PS: Jetwick is now free software … you can host your own and play around!