In modern tools of Computer Aided Design (CAD), NURBS and B-splines are a common mathematical way to describe bodies or structures. The geometric description yields smooth surfaces that share the property of higher order continuity and smoothness. Transferring the design to an engineering software or model, usually the properties of the design cannot be incorporated exactly within numerical computations. Therefore, Isogeometric Analysis (IGA) was introduced to combine geometry modelling and and the concept of finite elements by using B-splines and NURBS. However, complex domains require multiple and trimmed patches, i.e. the cutting away of dispensable parts is an important operation. Considering these features in an analysis is an issue as the definition of globally smooth basis functions is not straight forward. In this contribution, an approach for an scaled boundary isogeometric formulation (SB-IGA) for Kirchhoff--Love shells is presented that focuses on the handling of complex surfaces consisting of multi-patches including trimmed parts. More precisely, we divide our domain into a collection of star shaped blocks where each block is defined by a scaling of its boundary curve with respect to a specific point. Whereas often an analytical solution approach is chosen to solve the scaling direction, herein a discretized solution is chosen along the scaling direction using B-splines. Moreover, the patches within the block and across these blocks are coupled in a strong sense utilizing the analysis-suitable G^1 parametrization tailored for the scaled boundary ansatz. As a side effect we can handle trimming by considering the trimming curve as part of a modified boundary with the only requirement of blockwise star-convexity. By enforcing the higher order continuity of the basis functions across patch and block boundaries, the Kirchhoff--Love shell formulation can be implemented as it requires at least C^1 continuity of the test functions. The applicability and feasability of the presented procedure is demonstrated in several examples including complex and trimmed structures.