This paper explores operator growth and spread complexity in open quantum systems, focusing on the Sachdev-Ye-Kitaev (SYK) model. The authors introduce a measure of operator spread complexity based on the entropy of the population distribution of an operator over time. This measure is shown to be agnostic to the choice of operator basis and is effective in capturing the complexity of internal information dynamics in systems subject to an environment. The study demonstrates that the Krylov basis minimizes spread complexity, while the basis of operator strings is an eigenbasis for high dissipation. The paper also examines the long-time dynamics of the SYK model under decoherence and the effects of decoherence on complexity. The operator growth hypothesis (OGH) is used to define a quantum analogue of the classical Lyapunov exponent, which measures the rate of information spread in a system. The OGH has been successful in demonstrating linear growth of Lanczos coefficients for chaotic systems, both analytically and numerically. However, the converse is not necessarily true. The paper also discusses the use of the bi-Lanczos algorithm to generate a Krylov basis for open quantum systems, which is shown to be the minimal basis for describing the dynamics of a particular operator. The authors define an operator complexity measure based on the Shannon entropy of the population distribution of an operator. This measure is shown to be sensitive to the spread of the operator in a given basis and is used to study the onset of quantum chaos. The paper also shows that the Krylov basis minimizes the spread complexity, as demonstrated by comparing it with other bases. The study of the SYK model under decoherence shows that the Krylov complexity decreases with increasing decoherence strength, indicating that decoherence reduces the complexity of the dynamics. The paper concludes that the Krylov basis provides a natural choice for examining universal behaviour and growth rates in systems, while other bases, such as the Majorana string basis, are more suitable for understanding the dynamics in terms of operator size. The study highlights the importance of considering both information scrambling and decoherence in open quantum systems and suggests that future research should explore the effects of information backflow on the competition between scrambling and decoherence.This paper explores operator growth and spread complexity in open quantum systems, focusing on the Sachdev-Ye-Kitaev (SYK) model. The authors introduce a measure of operator spread complexity based on the entropy of the population distribution of an operator over time. This measure is shown to be agnostic to the choice of operator basis and is effective in capturing the complexity of internal information dynamics in systems subject to an environment. The study demonstrates that the Krylov basis minimizes spread complexity, while the basis of operator strings is an eigenbasis for high dissipation. The paper also examines the long-time dynamics of the SYK model under decoherence and the effects of decoherence on complexity. The operator growth hypothesis (OGH) is used to define a quantum analogue of the classical Lyapunov exponent, which measures the rate of information spread in a system. The OGH has been successful in demonstrating linear growth of Lanczos coefficients for chaotic systems, both analytically and numerically. However, the converse is not necessarily true. The paper also discusses the use of the bi-Lanczos algorithm to generate a Krylov basis for open quantum systems, which is shown to be the minimal basis for describing the dynamics of a particular operator. The authors define an operator complexity measure based on the Shannon entropy of the population distribution of an operator. This measure is shown to be sensitive to the spread of the operator in a given basis and is used to study the onset of quantum chaos. The paper also shows that the Krylov basis minimizes the spread complexity, as demonstrated by comparing it with other bases. The study of the SYK model under decoherence shows that the Krylov complexity decreases with increasing decoherence strength, indicating that decoherence reduces the complexity of the dynamics. The paper concludes that the Krylov basis provides a natural choice for examining universal behaviour and growth rates in systems, while other bases, such as the Majorana string basis, are more suitable for understanding the dynamics in terms of operator size. The study highlights the importance of considering both information scrambling and decoherence in open quantum systems and suggests that future research should explore the effects of information backflow on the competition between scrambling and decoherence.