Poisson regression (PR) is commonly used as the base model for analyzing count data with the restrictive equidispersion property. However, overdispersed nature of count data is very common in health sciences. In such cases, PR produces misleading inferences and hence give incorrect interpretations of the results. Mixed Poisson regression with individual--level random effects (MPR_ILRE) is a further improvement for analyzing such data. We compare MPR_ILRE with PR, quasi-Poisson regression (Q_PR) and negative binomial regression (NBR) for modelling overdispersed antenatal care (ANC) count data extracted from the latest Bangladesh Demographic and Health Survey (BDHS) 2014. MPR_ILRE is found to be the best choice because of its minimum Akaike information criterion (AIC) value and the overdispersion exists in data has also been modelled very well. Study findings reveal that on average, women attended less than three ANC visits and only 6.5\% women received the World Health Organization (WHO) recommended eight or more ANC visits for the safe pregnancy and child birth. Administrative division, place of residence, birth order, exposure of media, education, wealth index and body mass index (BMI) have significant impact on adequate ANC attendance of women to reducing pregnancy complications, maternal and child deaths in Bangladesh.