Paraphrase generation is important in various applications such as search, summarization, and question answering due to its ability to generate textual alternatives while keeping the overall meaning intact. Clinical paraphrase generation is especially vital in building patient-centric clinical decision support (CDS) applications where users are able to understand complex clinical jargons via easily comprehensible alternative paraphrases. This paper presents Neural Clinical Paraphrase Generation (NCPG), a novel approach that casts the task as a monolingual neural machine translation (NMT) problem. We propose an end-to-end neural network built on an attention-based bidirectional Recurrent Neural Network (RNN) architecture with an encoder-decoder framework to perform the task. Conventional bilingual NMT models mostly rely on word-level modeling and are often limited by out-of-vocabulary (OOV) issues. In contrast, we represent the source and target paraphrase pairs as character sequences to address this limitation. To the best of our knowledge, this is the first work that uses attention-based RNNs for clinical paraphrase generation and also proposes an end-to-end character-level modeling for this task. Extensive experiments on a large curated clinical paraphrase corpus show that the attention-based NCPG models achieve improvements of up to 5.2 BLEU points and 0.5 METEOR points over a non-attention based strong baseline for word-level modeling, whereas further gains of up to 6.1 BLEU points and 1.3 METEOR points are obtained by the character-level NCPG models over their word-level counterparts. Overall, our models demonstrate comparable performance relative to the state-of-the-art phrase-based non-neural models.