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In this study, we applied the ideas of subordination and the symmetric <i>q</i>-difference operator and then defined the novel class of bi-univalent functions of complex order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>. We used the Faber polynomial expansion method to determine the upper bounds for the functions belonging to the newly defined class of complex order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>. For the functions in the newly specified class, we further obtained coefficient bounds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="|" close="|"><msub><mi>ρ</mi><mn>2</mn></msub></mfenced></semantics></math></inline-formula> and the Fekete–Szegő problem <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="|" close="|"><msub><mi>ρ</mi><mn>3</mn></msub><mo>−</mo><msubsup><mi>ρ</mi><mrow><mn>2</mn></mrow><mn>2</mn></msubsup></mfenced></semantics></math></inline-formula>, both of which have been restricted by gap series. We demonstrate many applications of the symmetric Sălăgean <i>q</i>-differential operator using the Faber polynomial expansion technique. The findings in this paper generalize those from previous studies.