Lithium-ion batteries (LIBs) are widely used and became the main energy storage medium for many devices. Accurate estimation of LIBs state-of-health (SOH) is crucial for safe and reliable operation of devices. This study designs an end-to-end multi-battery shared hybrid neural network (NN) prognostic framework that combines a convolutional neural network (CNN), a multi-layer variant long-short-term memory (VLSTM) NN and a dimensional attention mechanism (CNN-VLSTM-DA) to SOH estimation for LIBs. First, feature extraction and selection on the raw input data are performed by using a CNN. Second, a suitable VLSTM is designed. The network adds a "peephole connection" to the forget gate and output gate, respectively, which enhances the network's ability to distinguish subtle features between input sequences. Besides, the forget gate and the input gate are coupled, so that, together, they determine the information that needs to be forgotten and the new data that needs to be added. Then, the output data of the CNN layer are fed into a multi-layer VLSTM NN to further capture the temporal correlation of these data. Finally, the attention mechanism is applied to the output of the VLSTM, to assign different weights to the features of each dimension and to give the prediction results. Several experiments are carried out on three datasets from NASA, CALCE and Oxford. These include full charge/discharge data, charge/discharge data in different SOC ranges, and non-fixed discharge current data. The results verify the effectiveness of the proposed method.