Cross-domain recommendation (CDR) has demonstrated to be an effective solution for alleviating the user cold-start issue. By leveraging rich user-item interactions available in a richly informative source domain, CDR could improve the recommendation performance for cold-start users in the target domain. Previous CDR approaches mostly adhere the Embedding and Mapping (EMCDR) paradigm, which learns a user-shared mapping function to transfer users' preference from the source domain to the target domain, neglecting users' personalized preference. Recent CDR approaches further leverage the meta-learning paradigm, considering the CDR task for each user independently and learning user-specific mapping functions for each user. However, they mostly learn representations for each user individually, which ignores the common preference between different users, neglecting valuable information for CDR. In addition, all these approaches usually summarize the user's preference into an overall representation, which can hardly capture the user's multi-interest preference. To this end, we propose a personalized multi-interest modeling framework for CDR to cold-start users, termed as NF-NPCDR. Specifically, we propose a personalized preference encoder that enhances the neural process (NP) with the normalizing flow (NF) to convert the Gaussian (unimodal) distribution to a multimodal distribution, providing a novel way to capture the user's personalized multi-interest preference. Then, we propose a common preference encoder with a preference pool to capture the common preference between different users. Furthermore, we introduce a stochastic adaptive decoder to incorporate both the personalized and common preference for cold-start users, adaptively modulating both preference for better recommendation.