Although advancing the therapeutic alternatives for treating deadly cancers has gained much attention globally, still the primary methods such as chemotherapy have significant downsides and low specificity. Most recently, Anticancer peptides (ACPs) have emerged as a potential alternative to therapeutic alternatives with much fewer negative side-effects. However, the identification of ACPs through wet-lab experiments is expensive and time-consuming. Hence, computational methods have emerged as viable alternatives. During the past few years, several computational ACP identification techniques using hand-engineered features have been proposed to solve this problem. In this study, we propose a new multi headed deep convolutional neural network model called ACP-MHCNN, for extracting and combining discriminative features from different information sources in an interactive way. Our model extracts sequence, physicochemical, and evolutionary based features for ACP identification using different numerical peptide representations while restraining parameter overhead. It is evident through rigorous experiments using cross-validation and independent-dataset that ACP-MHCNN outperforms other models for anticancer peptide identification by a substantial margin on our employed benchmarks. ACP-MHCNN outperforms state-of-the-art model by 6.3%, 8.6%, 3.7%, 4.0%, and 0.20 in terms of accuracy, sensitivity, specificity, precision, and MCC respectively. ACP-MHCNN and its relevant codes and datasets are publicly available at: https://github.com/mrzResearchArena/Anticancer-Peptides-CNN. ACP-MHCNN is also publicly available as an online predictor at: https://anticancer.pythonanywhere.com/.