Yes! Of course there could be, because there actually are such radically different moralities out there -- either actually being followed by some people, or at least being discussed as moral theories by philosophers.
The problem with understanding the intent of this question is fixing the referent for "our own". There are two ways of addressing this conundrum. One approach is to "smooth" or "average" the varieties of different moral codes that actual people do in fact seem to follow into a general approximate overall moral code that we can pretend all people follow. So "our own" becomes "our (all human beings) own". Then we can ask if there could be some morality that is radically different from this approximate generalization of a morality that we pretend all people (or maybe just moral people) follow. The other approach is to choose one set of people that pretend to adhere to one particular moral theory. Then we can ask if there could be some morality that is radically different from our (this particular set of people's) own. A variation on this second approach, is to choose the moral theory that I adhere to, and ask if there could be a morality that is radically different from my own.
Fortunately for a clear answer to the title question, it does not matter which approach is taken. No matter what concept of morality is used as the basis for the comparison, we can easily find a moral theory that is either actually practiced (or at least pretended to be practiced) or is discussed by philosophers that is as radically different as one wishes to require.
To give an example, consider two opposing moral theories that are as different from each other as it seems possible to imagine. One is the Christian morality, and the other is the morality defined by Evolutionary Ethics.
Christian morality is supposedly (self-avowedly, if not actually) practiced by a good proportion of the human race. It is an Divine Command / Authoritative Rule morality defined by the Ten Commandments of the Old Testament and the two commandments of Christ. (Various interpretations of Christianity add other Divine Commands gleaned from other parts of the Bible.) It is a non-consequentialist system of morality that determines the morality of an action (or person, etc.) on whether and how closely it follows the Divine commands. On some interpretations of Christ's second commandment ("Love thy neighbour as thyself") it is an altruistic morality in that it bestows more moral worth to any benefits that accrue to others than to any benefits that accrue to the acting agent. As a non-consequentialist system of morality, it maintains the moral-prudential divide. The command is "You shall love the Lord your God with all your heart, and with all your soul, and with all your mind." Circumstances and consequences are irrelevant to whether you obey it (although, of course, they may determine how you obey it). But the divine commands gleaned from the Bible are open to wide interpretation -- because the language is vague, the language is foreign (the original text is not in any modern language), the commands can often be interpreted to contradict each other, and the meaning of some of the words employed are uncertain given the context. So the Authoritative Rules involved in Christian morality are, to some extent, a matter of subjective interpretation -- the great variety of different Christian sects is proof of that.
Evolutionary Ethics, by comparison, is a morality that is practiced consciously by very few. Although the theory is that all people do in fact follow its basic principle, if for most only unconsciously and without rational consideration. It is a consequentialist system of morality, where the morality of an action is determined by how it best serves the inclusive fitness of the acting agent. It is, therefore, the exact opposite of an altruistic system of morality. And it is clearly an objective system of morality in contrast to Christianity's somewhat subjective nature. What debates and disagreements take place within an Evolutionary Ethics morality are about the facts of the matter and the science of population genetics -- whether a given action is or is not, or by how much, serving the acting agent's inclusive fitness.These two systems of morality are obviously radically different from each other -- on just about any axis of comparison. But if you don't like my choice of examples, the literature on moral theories is replete with similar radically different contrasts. Think of Egoistic Hedonism in contrast to a Marxian Altruistic morality. Or a Kantian deontological duty based morality in contrast to an Aristotelian virtue ethics morality. Depending on who is chosen as the referent for "our own" in the title question, it is easy to find a moral theory that is radically different from "our own", no matter how one interprets "radically different".
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