The paper explores the concept of operator growth and spread complexity in open quantum systems, focusing on the Sachdev-Ye-Kitaev (SYK) model. The authors introduce the *operator spread entropy* as a measure of complexity, which captures the dynamics of internal information in a system subject to an environment. This measure is agnostic to the choice of operator basis and is shown to be effective for the SYK model, both in its Krylov basis and the basis of operator strings. The Krylov basis, constructed using the bi-Lanczos algorithm, minimizes spread complexity, while the operator string basis is an eigenbasis for high dissipation. The authors demonstrate that decoherence reduces the complexity and Krylov space dimension, indicating a competition between information scrambling and leakage to the environment. They also show that the spread complexity in both bases captures the qualitative behavior of operator dynamics, providing insights into the interplay between scrambling and decoherence. The paper concludes by discussing the implications of these findings for understanding the competition between information scrambling and decoherence in open quantum systems, suggesting that the choice of basis can influence the interpretation of operator complexity.The paper explores the concept of operator growth and spread complexity in open quantum systems, focusing on the Sachdev-Ye-Kitaev (SYK) model. The authors introduce the *operator spread entropy* as a measure of complexity, which captures the dynamics of internal information in a system subject to an environment. This measure is agnostic to the choice of operator basis and is shown to be effective for the SYK model, both in its Krylov basis and the basis of operator strings. The Krylov basis, constructed using the bi-Lanczos algorithm, minimizes spread complexity, while the operator string basis is an eigenbasis for high dissipation. The authors demonstrate that decoherence reduces the complexity and Krylov space dimension, indicating a competition between information scrambling and leakage to the environment. They also show that the spread complexity in both bases captures the qualitative behavior of operator dynamics, providing insights into the interplay between scrambling and decoherence. The paper concludes by discussing the implications of these findings for understanding the competition between information scrambling and decoherence in open quantum systems, suggesting that the choice of basis can influence the interpretation of operator complexity.