The movement pattern you describe can be seen easier on a logarithmic chart:
(Chart courtesy of https://bitcoincharts.com/charts/bitstampUSD#rg60ztgSzm1g10zm2g25zv with BitStamp data because they have been around since 2011)
I have noticed a similar trend relating to the bitcoin halving. However, I think the pattern of the halving just so happens to line up well with fluctuating interest in the blockchain space. The early rally and recover periods between 2012 and end of 2014 had a lot to do with Silk Road and the technological flaws of bitcoin such as transaction malleability. Since then a lot of the price movement is related to government regulation and mainstream intrigue. The near future will probably follow similar reactions to mainstream adoption, as we can see recently with JPM coin and the Samsung Galaxy S10 hardware wallet. We may see a mooning in the next few years but it will be influenced by much more than just the next halving. :)
RE: Bitcoin tends to make new highs shortly after its block reward halving!