The pilot in the multi-engine airplane reported that the airplane touched down on the runway and she felt the application of aft yoke pressure. She looked over and saw that her right front seat passenger was "clenching the yoke with both fists." She reported that the airplane bounced once, and she aborted the landing. She flew one pattern and landed the airplane, and then she taxied to parking. According to the single-engine, pilot-rated passenger in the right front seat, during the initial touchdown, both engines remained at high rpm, and the airplane was drifting to the right side of the runway. The airplane bounced, and the pilot took the engines to idle. The airplane bounced again "very hard" on the nose landing gear. The airplane bounced a third time and landed "violently" on the nose landing gear. He reported that he felt the need to prevent a fourth runway impact, and he grabbed the yoke and applied aft pressure to minimize the rate of descent. The pilot in the left seat subsequently applied full power and aborted the landing. She flew one pattern and landed the airplane, then she taxied to parking. According to the passenger seated in the right rear seat, the approach was normal, but the initial touchdown was "forceful" and the airplane bounced. The airplane felt like it landed on the nose landing gear followed by the main landing gear and bounced a second time "much harder." A third bounce ensued, and it was the hardest. He heard the engines spool up and observed the pilot abort the landing. She flew one pattern and landed the airplane, and then she taxied to parking. The airplane sustained substantial damage to the fuselage, just below the windscreen. The pilot reported that there were no mechanical malfunctions or failures with the airplane that would have precluded normal operation.