<p>An extended (3+1)-dimensional Bogoyavl-ensky-Konopelchenko equation is the main focus of this study. We formulate the <i>N</i>th-order Pfaffian solutions using a combination of the Hirota method and Pfaffian technique, where <i>N</i> is a positive integer. The <i>N</i>th-order Pfaffian solutions are employed to determine the multi-soliton, multi-breather and hybrid solutions. Multi-soliton solutions reveal several interactions between the kink-shaped solitons, including their fusion and fission. Multi-breather solutions illustrate the fusion and fission of the two breathers, as well as the fusion and and fission of the one breather and two solitons. Elastic interaction between the one breather and one soliton is present through the hybrid solutions.</p>