With the rapid development of national defense and high-end equipment manufacturing industries of China, complex surface parts have been widely applied in fields like aerospace, transportation, and mold/die. Generally, the manufacturing of these parts relies heavily on computer-aided design (CAD) and computer-aided manufacturing (CAM) systems for modeling and process planning, and on computer numerical control (CNC) systems for final machining. As the key bridge of linking CAD, CAM, and CNC, tool paths not only directly determine how the machine tool moves, but also profoundly affects the machining efficiency of parts, so they play a decisive role in complex surface machining. However, traditional tool path planning methods mostly remain at the geometric level of achieving complete coverage of complex surfaces, making it difficult to directly consider the machining efficiency. To address this issue, a tool path planning method for complex surfaces with pointwise controllable machining strip width is proposed. First, by triangulating the complex surface and fitting local quadratic surfaces, an algorithm framework for calculating the maximal machining strip width direction field is established. On this basis, a breadth first search is employed to optimize the consistency and irrotationality of the initial field to ensure the regularity of the planned tool paths. Subsequently, the scalar field corresponding to the optimized direction field is obtained by solving a Poisson equation, and according to the quantitative correlation between the isolines of scalar field and the scallop height constraint, the isoline layout with pointwise control of the machining strip width is achieved on genus-0 surfaces. Furthermore, for nonzero genus surfaces, a tree-inspired segmented offsetting strategy for isoline layout is proposed by exploiting the local geometry anisotropy of surface near the genus to maximize the scallop height between adjacent isolines and thus optimize the total path length. Finally, a heuristic path connection strategy based on the shortest distance search is designed to reduce the air cutting distance among subregions caused by genus, ensuring further the overall path length. Simulation and real machining results indicate that, on genus-0 surfaces, compared with the traditional iso-scallop method and contour-parallel method, the proposed method reduces the total path length by up to 6.29% and 36.4%, respectively. Meanwhile, on nonzero genus surfaces, compared with the mainstream field-based tool path planning method, the proposed method can further shorten the total path length by more than 5%, and bring a 6.44% improvement in machining efficiency.