A friend of mine started having weird issues with his iPhone 4 after updating to 4.3. Touch screen lags, freeze ups, home button would cause apps to start shaking, and screen became unresponsive. Also, when clicking on the keyboard, sometimes the alternate letters would come up. He tried restoring the phone several times but still the problem remained. We stumbled onto a solution late into the 3rd hour.

Goto, Settings > General > Reset > (touch) Reset All Settings.

This leaves your data intact but resets the phone to defauts. Ringtones, email, sms preferences. etc..

Hope this saves you some time.